
Book 



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eOFiRIGKT DEPOSm 



GENERAL PHYSICS 



GENERAL PHYSICS 



AN ELEMENTARY TREATISE 
ON NATURAL PHILOSOPHY 



BY 

WILLIAM S. FRANKLIN 

AND 

BARRY MACNUTT 



A TEXT-BOOK FOR 
COLLEGES AND TECHNICAL SCHOOLS 



FIRST EDITION 



McGRAW-HILL BOOK COMPANY, Inc. 

239 West 39TH Street, New York 



LONDON: HILL PUBLISHING COMPANY, Ltd. 

6 AND 8 BOUVERIE ST., E. C. 
I916 



n 



Copyright, 191 6, by 
W. S. Franklin and Barry MacNutt 



/ 

MAR -I 1917 



PRESS OF 

THE NEW ERA PRINTING COMPANY 

LANCASTER, PA. 



3C!,A457268 



PREFACE. 

The teaching of elementary physics in our colleges and tech- 
nical schools has been growing more and more exacting for many 
years, and the ideal would seem to be an inescapable rigor, as 
in the teaching of mathematics, combined with a powerful 
imaginative stimulus such as can be realized only in the teaching 
of Natural Philosophy. We have prepared this elementary 
treatise on physics as a further step towards this ideal. 

A clear understanding of the two great generalizations of 
physics, namely, the Conservation of Energy and the Law of 
Entropy, would alone constitute a very considerable knowledge 
of Natural Philosophy, and we have therefore devoted a good 
deal of space to these things. See, particularly. Arts. 48, 80-81 
and 103-111. 

A clear understanding of Natural Philosophy is very greatly 
promoted by developing the contrasts between the methods of 
Mechanics, Thermodynamics, Atomics (the Atomic Theory) and 
Statistical Physics. The difference between Mechanics and 
Thermodynamics is set forth sufficiently by the difference be- 
tween Parts I and II; Thermodynamics and Atomics are con- 
trasted in Art. 79; Mechanics and Atomics are contrasted in 
Art. 227; and the student is referred to a very simple article on 
Statistical Physics in Science for August 4, 1916, pages 158-162. 

The Atomic theory is used but little in this text. We are 
sure that it would be a mistake to devote much space to the 
purely descriptive phases of the atomic theory in our treatment 
of heat; let the student go to Tyndall's book entitled Heat, 
A Mode of Motion. Likewise we believe that it would be in- 
compatible with the main purpose of this text to devote much 
space to the purely descriptive phases of the atomic theory of 
electricity and magnetism. An exacting elementary treatise on 



VI PREFACE. 

Natural Philosophy should, we believe, be limited almost exclu- 
sively to the methods of Mechanics and Thermodynamics as 
explained in Arts. 79 and 227. Therefore we have used the 
method of Thermodynamics in our treatment of heat, and the 
rest of the book, excepting only Chapter XVI, follows the method 
of Mechanics. The atomic theory is good medicine for the be- 
ginner in chemistry where it is used in connection with such 
precise ideas as atomic weight and valency; but the exacting 
and rigOiTOus use of the atomic theory is in nearly every other 
case beyond the mathematical powers of a beginner. 

We have been criticized by many of our mathematical friends 
for what they take to be our unfriendliness towards the exactions 
of mathematical teaching, but our unfriendliness goes no farther 
than to hold that to be exacting and unintelligible is the fatal 
error in teaching. Let no one imagine that we, in our unfriend- 
liness towards conventional mathematics teaching, fail to appre- 
ciate the necessity of the kind of precise thinking which is peculiar 
to the mathematical sciences, although much that has been said 
on this subject seems to us to be a sort of near-vision of that 
abstract heaven which, according to William James, is the only 
refuge of the tender-minded philosopher. 

W. S. Franklin and Barry MacNutt. 

South Bethlehem, Pennsylvania, 
October 24, 1916. 



TABLE OF CONTENTS. 
PART I. MECHANICS. 



PAGES 



CHAPTER 

I. Simple Dynamics 5-23 

II. The Arithmetic of Physics 24-58 

III. Friction. Work and Energy 59- 70 

IV. Hydrostatics 71-84 

V. Hydraulics 85-104 

PART II. THEORY OF HEAT. 

VI. Temperature and Thermal Expansion 107-118 

VII. The First Law of Thermodynamics 1 19-131 

VIII. Properties of Solids, Liquids and Gases 132-152 

IX. The Second Law of Thermodynamics 156-^^77 

PART III. ELECTRICITY AND MAGNETISM. 

X. Effects of the Electric Current 1 81-185 

XL Magnetic Effect of the Electric Current 186-213 

XII. Chemical Effect of the Electric Current 214-228 

XIII. Heating Effect of the Electric Current 229-251 

XIV. Induced Electromotive Force 252-280 

XV. Electric Charge and the Condenser 281-321 

XVI. Atomic Theory of Electricity 322-343 

PART IV. THEORY OF LIGHT. 

XVII. Light and Sound Defined 347-352 

XVIII. Some Useful Ideas of Wave Motion 353-358 

XIX. Regular Reflection and Refraction 359-370 

XX. Simple Lenses 371-380 

XXI. Simple Optical Instruments 381-391 

XXII. Lens Imperfections . . . .■ 392-418 

XXIII. Dispersion and Spectrum Analysis 419-430 

XXIV. Interference and Diffraction 431-441 

XXV. Polarization and Double Refraction 442-459 

vii 



Viu TABLE OF CONTENTS. 

CHAPTER PAGES 

XXVI. Photometry and Illumination 460-474 

XXVII. Color 475-484 

PART V. THEORY OF SOUND. 

XXVIII. Loudness, Pitch and Quality 487-493 

XXIX. Free Vibrations 494-518 

XXX. Forced Vibrations 519-523 

XXXI The Ear and Hearing 524-528 

APPENDIX A. PROBLEMS .- 529-592 

APPENDIX B. DIFFERENTIAL AND INTEGRAL 

CALCULUS 593-597 

INDEX 598-604 



PART I. 
MECHANICS. 

This treatise scarcely touches on the important topics of statics and elasticity. 
A very simple elementary treatise on statics with numerous problems, is Franklin 
and MacNutt's Elementary Statics, The Macmillan Co., 1915. A good intro- 
duction to the theory of elasticity is to be found in Franklin and MacNutt's Me- 
chanics and Heat, pages 183-218, The Macmillan Co., 1910. 

One of the most important topics in mechanics is wave motion. A simple intro- 
duction to this subject is to be found in Franklin, MacNutt and Charles' Calculus, 
pages 190-209, published by the authors, South Bethlehem, Pa., 1913. The 
possibilities of simple treatment of this difficult subject are illustrated in an article 
on Gun Recoil in The Journal of the Franklin Institute, Vol. 179, pages 559-572; 
spring, 191 5. 

The mathematical treatment of Electric Waves is identical to the mathemati- 
cal treatment of the simpler kinds of waves in material media, and the student of 
mechanics will find the chapter on waves (pages 193—273) in Franklin and Mac- 
Nutt's Advanced Electricity and Magneiism, The Macmillan Co., 1915, very helpful. 

A good book for the student is Alexander Ziwet's Theoretical Mechanics, The 
Macmillan Co., New York, 1894. 

A very good treatise on Hydromechanics, including Hydrostatics and Hydraulics, 
is Unwin's article Hydromechanics in the ninth edition of the Encyclopedia Britan- 
nica. 

More advanced books are: 

Treatise on Statics, G. M. Minchin, Oxford, 1884. 

Elementary and Advanced Rigid Dynamics (2 vols.), E. J. Routh, Macmillan & 

Co., London, 1892. 
Spinning Tops and Gyroscopic Motion, H. Crabtree, Longmans Green & Co., 

London, 1909. 
An extremely interesting discussion of the motion of a rigid body is Poinsot's 

Theorie Nouvelle de la Rotation des Corps, an old book which can be found 

in any good library. 
Mathematical Theory of Elasticity, W. J. Ibbetson, Macmillan & Co., London, 

1887. 
Treatise on the Theory of Elasticity, A. E. H. Love, Cambridge, 1892. 
Hydrodynamics, Horace Lamb, fourth edition, Cambridge, 1916. 
A recent development which has a vital bearing on mechanics is The Theory 

of Relativity; a good introductory monograph on this subject is Relativity 

and the Electron Theory, E. Cunningham, Longmans Green & Co., London, 

1915. 
2 I 



CALCULUS, THE ARITHMETIC OF PHYSICS. 

The method of differential ca^.culus is used quite freely in this 
elementary treatise on physics, but the authors believe that the 
use of this text does not depend upon the previous study of cal- 
culus by the student. Indeed the authors are convinced that 
the use of this book or its equivalent is a necessary preparat'on 
for the study of calculus; and the authors suggest that the student 
be required to turn again and again to the brief discussion of the 
methods of differential and integral calculus in Appendix B. 



INTRODUCTION TO MECHANICS. 

A great source of confusion In the study of physical science is 
the ambiguity of common words. Everyone knows, for example, 
that lead is heavier than cork, and yet the question "Which is 
heavier, a pound of lead or a pound of cork?" is unanswerable 
because the word heavy has two meanings. One pound of cork 
is heavier than half-a-pound of lead just as a ton of coal is 
heavier than half-a-ton of coal; in this case the word heavy refers 
to the amount of material as measured in pounds or grams. On 
the other hand, to say that lead is heavier than cork means 
that a piece of lead weighs more than an equal bulk of cork ; in 
this case the word heavy refers to an inherent property of cork 
and lead, and the word density is used to designate this inherent 
property of heaviness. Thus the density of water is about one 
gram per cubic centimeter or 62^ pounds per cubic foot, the 
density of lead is about 678 pounds per cubic foot, the density 
of cork is about 16 pounds per cubic foot, the density of kerosene 
is about 7 pounds per gallon. 

Everyone is familiar with the measurement of materials hy 
volume and hy weight, but everyone does not distinguish between 
the two methods in common use for measuring by weight, namely, 
(a) The method in which the spring scale is used, and (6) The 
method In which the balance scale is used. 

The spring scale measures the force with which the earth 
pulls on a body; but the force with which the earth pulls on a 
given body, as Indicated by a spring scale, Is slightly different 
at different places, and therefore the weight of a body as Indicated 
by a spring scale varies with location. 

The indication of the balance scale, on the other hand, does 
not vary with location, because the pull of gravity on the weighed 
body and the pull of gravity on the weights which are used to 
balance the body both change together. The result obtained by 

3 



4 MECHANICS. 

" weighing " a body on a balance scale is called the MASS of 
the body, and it is properly expressed in grams or pounds. 

It has been agreed to call the pull of the earth on a body the 
weight of the body. Thus the weight of a body, in the proper 
sense of the word, is a force, and is best expressed in dynes or 
poundals. The force with which the earth pulls on a one-pound 
body in London is a perfectly definite force and it is usually 
called the pound. Whenever the word pound is used in this 
sense in this text it is enclosed in quotation marks. Thus we 
will speak of lo pounds of sugar or 20 ''pounds" of force. 



CHAPTER I. 

SIMPLE DYNAMICS. 

I. Force* and its effects. — When one pushes or pulls on an 
object one is said to exert a force on the object. Likewise the 
pull of a horse on a wagon or the pull of a locomotive on a train 
is called a force. It does not, however, require an active agentf 
like a horse or locomotive to exert a force. Thus a weight lying 
on a table exerts a downward force (a downward push) on the 
table, a string stretched between two pegs exerts a force (a pull) 
on each peg. 

The effects of a force are extremely varied. Thus a horse 
must pull continuously on a wagon to overcome friction and keep 
the wagon moving steadily along a level road; to pull on a rubber 
band stretches it; to place a load on a beam bends or breaks the 
beam ; a coin is heated when a force is exerted upon it in rubbing 
it against a board; when steam is compressed it is condensed 
into water; when ice is compressed a portion of the ice melts; 
and so on. 

Another effect of a force is to change the velocity of a body. 
This is called the accelerating effect, and the study of this effect is 
called the science of dynamics. For example let us consider a 
falling body; if the body is a heavy metal ball falling only a 
few feet then the friction of the air is very small so that the 
only appreciable force acting on the ball is the downward pull 
of gravity, and the effect of this downward pull is to make the 

* There has been a great deal of discussion concerning the nature of force. A 
very interesting discussion of this and other fundamental matters in mechanics 
may be found in Ernst Mach's Mechanics, translated by Thomas J. McCormack, 
The Open Court Publishing Co., Chicago, 1907. Every serious student of me- 
chanics should read Mach's book. 

t There is an important difference between what is called an active force and 
what is called an inactive force. This matter is discussed in the chapter on Work 
and Energy. 

5 



6 MECHANICS. 

falling ball move more and more rapidly. As a second example 
let us consider a canal boat which is being brought up to full speed 
by the forward pull exerted by a mule ; in this case friction is not 
negligible. Let us suppose that the forward pull of the mule on 
the boat is loo "pounds" and that the backward drag of the 
water on the boat is 60 "pounds" at a given instant;* then at 
the given instant there is an unbalanced forward pull of 40 
"pounds" acting on the boat, and the effect of this unbalanced 
forward pull is to cause the velocity of the boat to increase at a 
definite rate. 

Constant and variable velocity. Constant and variable ac- 
celeration. — If a body travels over equal distances in equal 
times along a straight course or path the velocity of the body is 
said to be constant and it is equal to the quotient l/t, where / 
is the distance traveled in t seconds. If a body does not travel 
over equal distances in equal times along a straight path its 
velocity is not constant, and the value of the velocity at any 
particular instant is thought of as the limiting value of A//A/, 
where A/ is the distance traveled by the body during the very 
short interval of time A^. 

If a body gains equal amounts of velocity in equal times and 
if the gained velocity is always in the same direction the body is 
said to have a constant acceleration which is equal to the quotient 
v/t, where v is the velocity gained during t seconds. If a 
body gains unequal amounts of velocity in equal times or if the 
gained velocity is not always in the same direction then the 
acceleration of the body is not constant, and the value of the 
acceleration at a given instant is thought of as the limiting 
value of Av/At, where Av is the velocity gained by the body 
during the very short interval of time At. This instantaneous 
value of the acceleration of a body we will represent by the 
letter a. The acceleration of a body is the rate at which its velocity 
changes. Velocity and acceleration are discussed more at length 
in Chapter II. 

* The backward drag of the water grows larger and larger as the boat moves 
faster and faster. 




SIMPLE DYNAMICS. 7 

2. Balanced forces and unbalanced forces. — Figure i repre- 
sents two* equal and opposite forces A and B acting on a 
body. For example A may be 
the forward pull of a mule on a 
canal boat and B may be the 

backward pull of wind and water ^. 

^ Fig. I. 

on the boat. If A and B are 

equal and if they lie in the same straight line they balance each 
other, and the body behaves as if no force at all were acting on it.f 
When a single| force acts on a body we have what is called an 
unbalanced force. Thus, neglecting the friction of the air, a single 
force acts upon a falling body, namely the gravity pull of the 
earth. 

3. Newton's first law of motion. — When no force is acting on a 
body, or when the forces which do act are balanced, the body remains 
stationary, or, if moving, it continues to move with unchanging 
velocity along a straight path. Conversely, if a body is stationary 
or moving with unchanging velocity along a straight path then no 
force at all acts on the body or the forces which do act are balanced. 

Examples. — A book lies on a table. The earth pulls down- 
wards on the book and the table pushes upwards on the book with 
an equal force and the book remains stationary. If the table is in 
an elevator which is moving steadily along a. straight path, then 
the downward pull of the earth on the book and the upward push 
of the table on the book are equal and opposite (as before) but 
the book travels (with the elevator) along a straight path at un- 
changing velocity. 

When a horse draws a wagon at constant velocity along a 
straight road the forward pull of the horse on the wagon and the 
backward drag of the road on the wagon (friction) are equal. 

* The balanced relation of three or more forces is discussed in detail in the 
chapter on statics. This chapter is issued as a separate booklet, Elementary 
Statics, by Franklin and MacNutt, The Macmillan Co., price 50 cents. 

t Two equal and opposite forces applied at the ends of a rubber band produce 
a type of motion of the band which we call stretching. We are here concerned with 
the motion of a body as a whole. 

J This refers only to the simplest case. 



8 MECHANICS. 

The propeller shaft pushes forwards on a moving boat and the 
air and water drag backwards on the boat. When the boat is 
traveling steadily (at unchanging velocity) along a straight 
course, the backward drag of wind and water is exactly equal 
to the forward push of the propeller shaft. 

4. Newton's second law of motion. — When an unbalanced 
force acts on a body the velocity of the body changes either in 
value or in direction, or both. 

Examples. — The unbalanced pull of the earth on a falling 
body causes the velocity of the body to increase steadily in value. 
If the body is thrown straight upwards the unbalanced pull of the 
earth on the body causes its velocity to decrease steadily. 

A stone is tied to a string and twirled in a circle; the pull of 
the string on the stone is an unbalanced force and it causes the 
direction of motion of the stone to change continuously. 

A ball is tossed through the air, and the unbalanced pull of the 
earth on the ball causes the velocity of the ball to change in 
value and in direction. 

Consider a body upon which an unbalanced force F is acting 
as shown in Fig. 2. After the force has acted for M seconds the 
body will have gained a certain amount of velocity Az; in the 
direction of F. If the body has no velocity at the beginning, 



body body 




Fig. 2. Fig. 3. 

then its actual velocity at the end of the time M will be what it 
has gained, namely, Az;. But if the body has a certain velocity 
V at the beginning, then its actual velocity at the end of the time 



SIMPLE DYNAMICS. 9 

A/ will be what is called the vector sum* of v and Av as shown 
in Fig. 3. 

The gain of velocity Av during a given interval At is propor- 
tional to F and inversely proportional to the mass m of the body, or, 

Av 
The rate of gatn of velocity, — ,, that is the acceleration, a, of 

the body, is proportional to F and inversely proportional to m. 

5. Dynamic units of force. The dyne and the poundal. — We 

may choose as our unit of force that force which acting as an 
unbalanced force on a body of unit mass will produce unit 
acceleration. 

Thus the dyne is the force which will produce an acceleration of 
one centimeter per second per secondf when acting as an un- 
balanced force on a one-gram body. The pull of the earth on a 
one-gram body produces an acceleration of about 980^ centi- 
meters per second per second. Therefore the pull of the earth 
on a one-gram body is about 980 dynes. 

The poundal is the force which will produce an acceleration of 
one foot per second per second when acting as an unbalanced 
force on a one-pound body. The pull of the earth on a one- 
pound body in London produces an acceleration of 32.16 feet 
per second per second and this force is therefore equal to 32.16 
poundals. 

6. Formulation of the second law of motion. — The accelera- 
tion a of a body is proportional to the accelerating force F 
and inversely proportional to the mass m of the accelerated body 
according to Art. 4. Furthermore we have chosen our unit 
force as that force which will produce un;it acceleration when 

* Every student of High School physics knows what is meant by the " parallel- 
ogram of forces"; it is the scheme for finding what is called the vector sum of two 
forces. Velocities are added in the same way as forces. See Art. 20 of Chapter 
II. 

t Acceleration is so much velocity gained per second, so many centimeters per 
second of velocity gained per second, or, briefly, so many centimeters per second 

PER SECOND. 

% Varying from about 978 centimeters per second per second at the equator to 
about 982 centimeters per second per second near the poles of the earth. 



10 MECHANICS. 

acting (as an unbalanced force) on a body of unit mass. There- 
fore F units of force will produce F units of acceleration when 
acting on a body of unit mass (acceleration proportional to force), 
or Fjm units of acceleration when acting on a body of which 
the mass is m units (acceleration inversely proportional to mass 
of body). Therefore, representing the acceleration Fjin by a, 
we have: 

F 

— = a 

m 

or 

F = ma (i) 

in which F is the value of an unbalanced force in dynes (or 
poundals), m is the mass of the accelerated body in grams (or 
pounds) , and a is the acceleration in centimeters per second per 
second (or feet per second per second). 

If we use the pull of the earth on a one-pound body at London 
as our unit of force and call it a "pound" for brevity; then one 
"pound" of force will produce an acceleration of 32.16 feet per 
second per second when acting on a one-pound body, / "pounds" 
of force will produce / times as much acceleration when acting 
on a one-pound body, or f/m times as much acceleration when 
acting on an m-pound body. That is, 



whence 



/ 
a = ~X 32.16 

m 



f = ~ ma (2) 

32.16 



where / Is the value of an unbalanced force in "pounds," m 
is the mass in pounds of the body upon which the force acts, 
and a is the acceleration in feet per second per second produced. 
Many engineering writers call the fraction w/32.16 the mass 
m' of the body so that equation (2) becomes 

/ = m'a (3) 



SIMPLE DYNAMICS. II 

The quantity m' is the mass of the body expressed in terms of 
a unit 32.16 times as large as the pound. This is sometimes 
called the gee-pound or the slug. A gee-pound is an amount of 
material which is accelerated at the rate of one foot per second 
per second by an unbalanced force of one "pound" (the pull of 
the earth on a one-pound body in London) . The use of equa- 
tion (j) must not lead one to lose sight of the fact that mass is properly 
and almost universally expressed in pounds and very rarely 
expressed in slugs. 

It has been agreed that the technical meaning of the word 
weight shall be the force with which the earth pulls a body, whereas 
in everyday life the word weight nearly always means exactly 
what it has been agreed to call mass. A ton of coal measured 
out by a balance scale is 2,000 pounds mass of coal. 

The pull of the earth on an m-pound body produces an accelera- 
tion of, say, g feet per second per second, and therefore the pull 
of the earth on the m-pound body (the weight W of the body) 
is mg poundals according to equation (i). That is 

W = mg (4) 

The value of g varies from place to place, and of course W 
varies also, since m is invariable. 

From equation (4) it is evident that the mass w of a body is 
equal to its weight W divided by the acceleration of gravity g, 
but it is not correct to call 2,000 pounds of coal (as measured 
by a balance) a WEIGHT of coal and divide by g to get the 
MASS of the coal. 

7. Action and reaction. Newton*s third law of motion. — 

When one pushes against an object with the hand, one can feel 
a back-push on the hand; the forward push of the hand on the 
object necessarily involves a backward push of the object on the 
hand, and these two opposite pushes are equal in value. The 
force action between two bodies A and B always consists of a 
pair of equal and opposite forces {action and reaction they are 
called) one of which acts on A and the other on B. 



12 



MECHANICS. 



As an example of action and reaction consider a book resting 
on a table; the book pushes downwards on the table and the 
table pushes upwards on the book with an equal force. Or 
consider a locomotive drawing a car as shown in Fig. 4. At the 
point h there is force action, the locomotive pulls forwards on 



o 



locomotive 







a c c 

Fig. 4. 

the car and the car pulls backwards with an equal force on the 
locomotive. At aa there is force action, the rims of the drive 
wheels push backwards on the rails and the rails push forwards 
with an equal force on the rims of the drive wheels. At cc 
there is force action, the track drags backwards on the rims of 
the car wheels (because of friction) and the rims of the car 
wheels pull forwards on the rails with an equal force. 

It must be clearly understood that action and reaction are 
exerted on different bodies. Thus the pair of forces at a in 
Fig. 4 consists of a forward pull on the car, and an equal back- 
ward pull on the locomotive, and this pair of forces constitute 
action and react on. . To make this matter perfectly clear let us 
consider what forces are acting on the car in Fig. 4, namely, 
a forward pull on the car at h (this force is exerted by the 
locomotive), and a backward drag exerted on the car by the 
rails at cc. These two forces exerted on the car are not equal, 
that is, they do not balance each other, when the speed of the 
car is changing (see Newton's second law of motion) ; but these 
two forces exerted on the car are equal, that is, they do balance 
each other, when the speed of the car is not changing. 

Newton's third law of motion, the equality of action and re- 



SIMPLE DYNAMICS. 13 

action, refers to a very simple and familiar fact, so simple and 
familiar that many students overlook it entirely, and in many 
cases the student is led astray by false examples such as the 
following: "Consider the force with which a mule pulls forwards 
on a rope and the force with which a canal boat pulls backwards 
on the rope. These two forces are equal and opposite, the mass 
of the rope being negligible.''' The two forces referred to do not 
constitute action and reaction in the precise sense in which these 
words are used by Newton, and they are not necessarily equal 
and opposite as the author from whom we have quoted admits 
when he assumes the mass of the rope to be negligible! The 
truth of Newton's third law is not conditioned upon negligible 
mass by any means ! 

8. Translatory motion and rotatory motion. — Motion in which 
every line in a body remains unchanged in direction is called 
translatory motion. The motion of a body along a straight path 
Is sometimes called straight line motion. Thus the motion of a 
car along a straight track and the motion of a boat along a 
straight course are examples of straight line motion. Straight 
line motion is, of course, translatory motion, but it is translatory 
motion of a very simple type. The most general case of trans- 
latory motion is as follows : Grasp 

^-center of mass . 

a long slim stick at its middle ^^^==:^^=^_ ^=^-^^=^ ^^=^ ^^ 

and move the hand in any Irregu- / 

lar way whatever, as suggested ^^l>.^^^Si^i^S.2.iBiit^ 

by the dotted curve In Fig. 5, « — — — j^- * 

but without turning the stick in "^ 

any way; that is, if the stick is . ^ 

initially vertical let it remain 

vertical during its entire motion, and if a certain edge of the stick 

is due north and south let it remain due north and south during 

the entire motion. 

Motion In which a certain line in the body remains stationary 
is called rotatory motion, and this stationary line Is called the 
axis of rotation. 



14 MECHANICS. 

Rotation about a fixed axis is a simple special case of rotatory motion. Imagine 
a certain point in a body to be fixed or immovable, the center point of a sphere 
for example, then if the body is turned irregularly about the fixed point we have the 
most general case of rotation. In this case the body may be at one moment turning 
about one axis and at another moment turning about another axis; both axes 
must however pass through the fixed point. 

Every case of actual motion involves translatory motion, and 
rotatoty motion, and extremely complicated changes of size and 
shape all mixed up together; and one may well ask "What is 
the use of discussing the ideally simple types of motion, trans- 
latory motion and rotatory motion?" Because it is necessary 
to discuss one thing at a time, and because in a great many im- 
portant cases the extreme complications are negligible. Thus, 
for most practical purposes, one may think of the motion of a 
railway train on a straight track as a simple case of translatory 
motion, whereas the actual motion involves the swaying and 
vibration of the cars, and the rattling of every loose part, it 
involves a complicated movement of oil and of wearing metal in 
the journals, it involves a yielding of track and ballast, and it 
involves a whirling and eddying motion of the surrounding air 
as may be seen in the infinitely complicated motion of dust 
and smoke. 

9. Center of mass. — ^Take hold of a long slim stick, at its 
middle, between thumb and finger, and move the stick up and 
down and to and fro in any manner but without turning the stick. 
Closing one's eyes it seems as if one had a heavy piece of lead 
between thumb and finger, that is to say, the material of the 
stick seems to be concentrated between thumb and finger, and 
this point is called the center of mass of the stick — but if one twirls 
the stick even to a very slight extent it no longer feels like a 
piece of lead between thumb and finger! This experiment must 
be tried to be appreciated. 

The center of mass of a body is the point at which all of the 
material of the body may be thought of as concentrated in so 
far as the behavior of the body is concerned when it is performing 
translatory motion; or the center of mass of a body is the point 



SIMPLE DYNAMICS. ' 15 

at which a single force must act on the body to produce trans- 
latory motion. 

To observe the behavior of a stick when acted on by a single 
force which is not applied at the center of mass of the stick 
one may strike the stick with a hammer as indicated in Fig. 7. 



center of mass 



hammer hammer 

^^ center of mass -— V-T^ 

\r '''"'" 

Fig. 6. Fig. 7. 

The force exerted by the hammer is so large, while it lasts, that 
the pull of gravity on the stick may he ignored. The stick is to be 
held horizontally by the hand and released at the instant of the 
hammer blow. When a sudden force is exerted at the center of 
mass of the stick as indicated in Fig. 6 the stick is set in trans- 
latory motion without rotation. When a sudden force is applied 
not at the center of mass as indicated in Fig. 7, a combination 
of translatory and rotatory motion is produced.* 

In the discussion of Newton's laws of motion in Arts. 3-7 
translatory motion only is considered. That is to say, the force 
which acts on the body under consideration is supposed to be 
applied at the center of mass of the body. 

10. Rotation about a fixed axis. Definitions. — The simplest 
case of rotatory motion is exemplified by the rotation of a wheel 
about a fixed axis. 

Spin velocity. — Let </> be the angle turned by a rotating 
body in t seconds. Then the quotient 4)lt is the average rate 
of turning of the body or the average spin velocity of the body 
during the time t. Spin velocity may of course be expressed in 
degrees of angle per second, but in practice it is usually expressed 

* The difference between Figs. 6 and 7 may be made evident to an entire class 
by calling the attention of the class to the sound produced when the stick strikes 
the floor. In Fig. 6 the stick strikes the floor flat-wise and produces a sharp slam. 
In Fig. 7 one end of the stick strikes the floor and a clattering sound is produced. 



1 6 MECHANICS. 

in revolutions per second. Spin velocity is represented by the 
letter s in this text, and it is understood to be expressed in 
radians* per second. There are 27r radians in one revolution, 
and therefore n revolutions per second is equal to 27rn radians 
per second. That is 

s = 2Tn (5) 

where n is the spin velocity of a body in revolutions per second 
and s is its spin velocity in radians per second. 

Spin acceleration. — In raany machines a part may rotate at 
a variable spin velocity. Thus the balance wheel of a watch 
gains and loses spin velocity repeatedly. In the study of such 
motion it is necessary to consider the rate of gain (or loss) of 
spin velocity, just as It Is necessary to consider the rate of gain 
(or loss) of velocity (the acceleration) of a body in translatory 
motion. The rate of change of the spin velocity of a body is 
called Its spin acceleration. As an example consider an electric 
motor which Is started from standstill and reaches a speed of 
20 revolutions per second in 6 seconds. The average rate of 
gain of speed (average spin acceleration) during the six seconds Is 
20 revolutions per second divided by 6 seconds, or 3.33 revolu- 
tions per second gained per second, or, simply, 3.33 revolutions 
per second per second. To reduce to radians per second per 
second we must multiply by 27r because there are 27r radians 
in one revolution, so that 3.33 revolutions per second per second 
is 10.47 radians per second per second. 

Turning force or torque. — In spinning a simple top by thumb 
and finger It is necessary to exert equal and opposite forces by 

* Consider the angle between two intersecting straight hnes. Imagine a 
circle of radius r drawn with its center at the apex of the angle (point of inter- 
section of the lines), and let a be the length of the arc of the circle which lies be- 
tween the lines which bound the angle 4>. Then the ratio ajr (arc divided by 
radius) has a fixed value for the given angle, and the value of this ratio expresses 
the value of the angle in radians, the radian being the angle for which a = r. 
The entire circumference of a circle is equal to 2Trr, and circumference divided by 
r gives 2ir, so that there are 27r radians in the angle corresponding to an entire 
circumference. 



SIMPLE DYNAMICS. 



17 



thumb and finger as indicated by the arrows in Fig. 8, otherwise 
the top is thrown to one side (set into translatory motion). 
When the forces exerted by thumb and finger are equal and 
opposite, however, a simple spinning motion is produced. Simi- 





upward push of hearing 
shaft of grindstone 



rod 




Fig. 9. 

larly, two equal and opposite forces must be exerted on an auger 
handle as shown in Fig. 9. A pair of equal and opposite forces 
exerted in this way (the two forces not lying in same straight 
line) constitutes what is called a turning force or torque. 

In turning a grindstone a single force 
B is exerted on the grind-stone crank as 
shown in Fig. 10, but the bearing in 
which the grindstone shaft rotates exerts 
on the shaft a force A which is equal 
and opposite to B as shown in the crank- 
figure, and the pure rotatory motion of 
the grindstone is produced by the com- 
bined action of A and B, 

The single force B in Fig. 10 (like 
the single force due to the hammer 
blow in Fig. 7) would, if acting alone, produce a combination of 
translatory motion and rotatory motion. Nevertheless we fre- 
quently speak of the turning or torque action of a single force 
about a fixed axis, because, when a body is fixed on an axis 
(or shaft) the force at the axis (like A in Fig. 10) is exerted by the 
bearing automatically and we ordinarily do not think about it. 

II. The principle of moments. Balanced torque actions. — ■ 
A lever is supported on a fixed axis at (axis perpendicular to 
3 



B 



Fig. 10. 



I 



i8 



MECHANICS. 



plane of paper) as shown in Fig. ii, and two forces A and B 
act on the lever. In order that the lever may not he set rotating 

lever 




Y 
A 



'^■ 



Fig. II. 



about the axis the torque actions of the forces A and B about 
the axis must be equal and opposite, and this condition is expressed 
algebraically by the equation : 

Aa = Bb (6) 

where a and b arethedistancesshowninFig.il. This equa- 
tion may be easily verified experimentally, and in accordance 
with this equation we may take the product Aa as an expression 
for the torque action of the force A about the axis (this torque 
tends to turn the lever in a counter-clockwise direction about 0), 
and we may take the product Bh as an expression for the torque 
action of the force B about the axis (this torque tends to 
turn the lever in a clockwise direction about 0). 

Note. — The value of a torque is expressed as the product of a 
force and a distance (length of lever arm). Therefore the unit 
of torque is the dyne-centimeter (or the poundal-foot) . 

12. Newton's laws of motion as applied to the rotation of a 
perfectly symmetrical body like a ball.* First law. — A rotating 

* The discussion of the rotatory motion of a non-symmetrical body is beyond 
the scope of this text. 

Poinsot's Theorie Nouvelle de la Rotation des Corps is, perhaps, the most intel- 
hgible account of the rotation of a non-symmetrical body. Spinning Tops and 
Gyroscopic Motion by H. Crabtree (Longmans, Green & Co., 1909) is the best 
simple treatise. The most complete treatise on the motion of a rigid body is 
Advanced Rigid Dynamics by E. J. Routh, Macmillan & Company, London, 1902. 



SIMPLE DYNAMICS. 19 

symmetrical body remains at rest or continues to rotate at a 
constant spin velocity about a fixed-direction axis if no torque 
acts upon it or if the torques which do act are balanced. Thus 
when a grindstone is kept steadily rotating, the retarding torque 
due to friction is balanced by the driving torque. A spinning 
base ball continues to spin about a fixed-direction axis except 
insofar as the spinning motion is modified by torque action 
exerted on the ball by the surrounding air. 

Second Law. — ^When an unbalanced torque acts upon a rotating 
symmetrical body it causes the spin velocity to change in value, 
or the direction of the axis of spin to change, or both. 

Before undertaking a precise discussion of the second law of 
motion as applied to a rotating symmetrical body it is necessary 
to consider a method for representing spin and torque by lines 
in a diagram. 

Figure 12 represents a spinning wheel. To represent the spin 
of the wheel by a line, draw the line ^ in the direction of the 




(^^ fe)_£-*. 




Fig. 12. 

axis of spin and place the arrowhead on that end of ^ towards 
which the spinning wheel would travel if it were a nut turning 
on a right-handed screw thread. 

Figure 13 represents a torque acting on a screw driver. To 
represent the torque by a line, draw the line T in the direction 
of the axis of the torque (axis of screw driver) and place the arrow- 
head on that end of T towards which the torque would cause a 
nut to travel on a right-handed screw thread. 

Consider a symmetrical body which is acted upon by an 
unbalanced torque as indicated by the arrow T in Fig. 14a. 



20 



MECHANICS. 



After the torque has acted for a certain time M the ball will 
have gained a certain amount of spin velocity As about the axis 
of the torque as represented by the arrow As in Fig. 146. If 



hall 




^^axis of torque 

Fig. 14a. 




'^actual spin 
after time 6t 



spin produced by T 
during time At 

Fig. 14b. 



the ball has no spin at the beginning then its actual spin at the 
end of the time interval A/ will be what it has gained, namely, 
A5; but if the ball has at the beginning a certain amount of spin 
5 as represented by the arrow 5 in Fig. 14&, then the actual spin 
of the ball at the end of the time interval A^ will be what is 
called the vector sum of 5 and A5 as shown in Fig. 14&. That is 
to say, after the time interval M the ball will be rotating about 
OB as an axis at a speed which is represented by the length of 
OB. 

A symmetrical body when acted upon by an unbalanced torque 
gains spin velocity about the axis of the torque at a rate (As/A/) 
which is proportional to the torque and inversely proportional to 
what may be called spin-inertia of the body. 

13. Explanation of spin-inertia. — The property of a body by 
virtue of which it remains at rest or continues to move steadily 
along a straight path when not acted upon by an unbalanced 
force is called the inertia of a body ; but the meaning of the word 
inertia is generally extended to indued the idea of reluctance to 
gain velocity. A body of large mass must be acted upon by a 
large force to give it a certain amount of acceleration, indeed 
the greater the mass of a body the greater its reluctance to gain 



SIMPLE DYNAMICS. 



21 



velocity. Therefore the mass of a body Is a measure of Its 
reluctance to gain velocity, that Is, of its inertia. 

Similarly, the reluctance of a body to gain or lose spin velocity 
may be called Its spin-inertia. 

Experiment with a slim stick. — The spin-inertia of a given 
body varies with the position of the axis of rotation. Thus a 
slim stick may be very easily set spinning about the axis 00 
in Fig. 15, less easily about the axis 00 in Fig. 16, and still less 
easily about the axis 00 in Fig. 17. That Is to say, a given 



O 



Q slim stick q 



slim stick 



O 



O 



slim stick 







Fig. 15. 



Fis. 16. 



Fig. 17. 



torque would produce a given spin velocity in a very short time 
about the axis 00 In Fig. 15, the same torque would have to 
act much longer to produce the same spin velocity about the 
axis 00 In Fig. 16, and It would have to act still longer to produce 
the same spin velocity about the axis 00 in Fig. 17. These 
statements are easily verified by trial. Therefore the spin-inertia 
of the stick is small about the axis 00 In Fig. 15, it Is larger 
about the axis 00 in Fig. 16, and still larger about the axis 00 
in Fig. 17. 

14. Definition of spin-inertia.* Formulation of second law of 
motion as applied to the rotation of a symmetrical body. — From 
equation (i) of Art. 6 we have m = F/a, that is, the mass or 
inertia of a body is equal to the quotient F/a, where F is an 
unbalanced force acting on the body and a Is the acceleration 
it produces. Similarly, let T be an unbalanced torque (about 

* Spin-inertia is frequently called moment of inertia. 



22 



MECHANICS. 



a given axis) acting on a body and let a be the spin acceleration 
it produces. Then the quotient Tla may be defined as the 
spin-inertia of the body about the given axis. Therefore, using 
the letter K for the spin-inertia of the body about the given 
axis, we have 



T 



or 



T = K 



a 



(7) 



In this equation a is the spin acceleration of the body in radians 
per second per second, T is the unbalanced torque in dyne- 
centimeters (or poundal-feet) , and K is the spin-inertia of the 
body about the axis of the torque. Spin-inertia is expressed 
as the product of a mass and the square of a distance, namely, 
gram- (centimeter)^ [or pound- (foot) 2]. 



TABLE. 
Values of Spin-Inertia for Some Regular Homogeneous Solids* 



Shape of solid and position of axis. 



Value of J^. 



Solid sphere of radius r and mass m, axis of rotation passing 
through center of sphere 



SoHd cylinder of radius r and mass m, axis of rotation same 
as axis of cylinder 



Very slim rod of length I and mass m, axis of rotation at right 
angles to rod and passing through center of rod 



Rectangular parallelpiped of length I and breadth b and 
mass m, axis of rotation at right angles to / and b and 
passing through center of parallelpiped 



f mr^ 



^ mr^ 



T^ 



t\ w(/2 + &2) 



15. Newton's third law of motion as applied to torque action. 

— If one body A exerts a torque upon another body B, then 
body B exerts an equal and opposite torque on A. 

* These values are calculated by the method of integral calculus. See Appen- 
dix S. 



SIMPLE DYNAMICS. 



23 



UNIFORMLY ACCELERATED MOTION. 

16. Falling bodies. — When a constant* unbalanced force acts 
on a body the body gains velocity at a constant* rate. Such a 
body is said to perform uniformly accelerated motion. A body 
which is falling freely under the action of the constant pull of 
gravity is, in so far as the friction of the air is negligible, an ex- 
ample of uniformly accelerated motion. If the body is moving 
vertically we have the simple falling body, if the body is thrown 
like a ball we have what is called a projectile. 

Let g be the constant rate of gain of velocity by a falling 
body, then gt is the amount of downward velocity gained in t 
seconds. That is 

gain of velocity = gt (i) 

Let Vi be the velocity of the falling body at the beginning of 
the / seconds, then Vi -\- gt is the velocity at the end of the 
t seconds. 

Now the average value during any given time of any quantity 
which changes at a constant rate is equal to half the sum of the 
values at the beginning and 
end of the time,t so that the 
average velocity of a falling 
body during t seconds is half 
the sum of Vi and v^ + gt. 
That is, the average velocity 
is Vi + ^gt, and the distance 
d traveled by the falling body 
is equal to the average velocity multiplied by the time. Therefore : 

d = v,t + \gt^ (ii) 

If the velocity Vi at the beginning is upwards it is to be con- 
sidered as negative, and d is positive or negative according as 
the body is below or above its starting point after t seconds. 

* Constant in value and unchanging in direction. 

t This proposition may be understood as follows: Let the constantly increasing 
velocity of a falling body be represented by the ordinates of a curve as shown 
in Fig. 18. This "curve" is a straight line, and the average ordinate of any portion 
AB of this straight line is equal to ^{vi + vi). 




axis of time 

Fig. 18. 



■■■■■■■■■■MMiMi 



CHAPTER II. 

THE ARITHMETIC OF PHYSICS. EXEMPLIFIED BY A FURTHER 
DISCUSSION OF TRANSLATORY AND ROTATORY MOTION. 

17. Measures and units of quantity. — In the expression of a 
physical quantity two factors always occur, a numerical factor 
and a unit, and the numerical factor is called the measure of the 
quantity. Thus we speak of a length of 67 centimeters, a force 
of 265 dynes, a time of 25 seconds, etc. 

It is a great help towards a clear understanding of physical 
calculations to consider that units and meastires are both involved 
when one physical quantity is multiplied by or divided by an- 
other. Thus a rectangle is 50 centimeters wide and 100 centi- 
meters long, and its area is 50 centimeters multiplied by 100 
centimeters, or 5,000 square centimeters. A train travels 250 
miles in five hours, and its average velocity is 250 miles divided 
by 5 hours, or 50 miles per hour. The length a of the lever arm 
in Fig. II is 3 feet, the force A is 320 poundals and the torque 
action Aa of the force about the axis is 960 foot-poundals. 
A solid sphere having a radius of 2 feet and a mass of 7,500 
pounds has a spin-inertia of 3,000 pounds X (2 feet)^ or 12,000 
pound-feet-square according to the table in Art. 14. 

Products and quotients of units arrived at in this way are new 
physical units. Thus the centimeter squared is a unit of area, 
the mile per hour is a unit of velocity, the foot-poundal or the 
centimeter -dyne is a unit of torque, the pound-(foot)2 or the gram- 
(centimeter)^ is a unit of spin-inertia. The word per connecting 
the names of two units indicates that the following unit is a 

divisor. Thus the mile per hour is frequently written r 

. . . . cm. 
and the centimeter per second is frequently written . A 

24 



THE ARITHMETIC OF PHYSICS. 25 

hyphen connecting two units indicates that the two units are 
multipHed together. 

It is important to carry the units through with every numerical 
calculation, the arithmetical operations among the various units being 
indicated algebraically. When this is done there can be no 
ambiguity as to the meaning of the result. When this is not 
done the result has, strictly speaking, no physical meaning at all. 

Although the unit in terms of which a result is expressed is 
known when the units are carried through a numerical calcula- 
tion, it frequently happens that the unit is so entirely unfamiliar 
that it might almost as well be unknown. Thus the rule for 
finding the area of a rectangle by taking the product of length 
and breadth is entirely general no matter what units of length 
and breadth are used, and a rectangle 2 feet long and 6 inches 
wide has an area of 12 foot-inches! Now the foot-inch is the 
area of a rectangle one foot long and one inch wide but it is 
entirely unfamiliar as a unit of area so that one might almost 
as well not know the value of an area at all as to have it given 
in terms of such a unit! It is, for this reason, nearly always 
necessary to simplify the data of a problem before the data 
can be used intelligibly in numerical calculations. Thus it may 
be required to calculate the force which will produce an accelera- 
tion of 20 feet per second per second when acting on a body of 
which the mass is 10 kilograms, using equation (i) of Art. 6. 
Before this equation can be used, however, the acceleration must 
be expressed in centimeters per second per second and the mass 
must be expressed in grams (then the force will be given in dynes) ; 
or the acceleration must be expressed in feet per second per second 
and the mass must be expressed in pounds (then the force will 
be given in poundals). In equation (2) of Art. 6 mass must 
be expressed in pounds and acceleration in feet per second per 
second, and the value of the force is then given in " pounds " as 
explained in Art. 6. 

18. Systems of units. — A set of units which can be used to- 
gether most simply in numerical calculations is called a system 



26 



MECHANICS. 



of units. There are two important systems of units, namely, the 
centimeter-gram-second system (the c.g.s. system) and the foot- 
pound-second system (the f.p.s. system). Following is a partial 
list of the units in these two systems. 



Quantity. 


C.G.S. Unit. 


F.P.S. Unit. 


length 


centimeter 


foot 


mass 


gram 


pound 


time 


second 


second 


area 


(cm.)2 


(f00t)2 


volume 


(cm.)3 


(f00t)3 


velocity 


cm. per sec. 


foot per sec. 


acceleration 


cm. per sec. per sec. 


foot per sec. per sec. 


spin-velocity 


radian per sec. 


radian per sec. 


spin-acceleration 


radian per sec. per sec. 


radian per sec. per s( 


force 


dyne 


poundal 


torque 


dyne-centimeter 


poundal-foot 


spin-inertia 


gram-(cm.)2 


pound-(foot)2 


work or energy 


erg 


foot-poundal 


etc. 


etc. 


etc. 



Every equation in this text, except equation (2) of Art. 6 and 
equation (27) of Art. 53, can be used with c.g.s. units or with 
f.p.s. units ; but the units of the two systems must not be mixed. 

19. Scalar quantity and vector quantity. — A scalar quantity is 
one which does not have direction but has magnitude only. 
Thus volume, mass, time, energy are scalar quantities; and it is 
a complete specification to say 10 cubic meters of sand, or 2 
pounds of sugar, or 5 seconds of time, or 2,000 units of energy. 

A vector quantity is one which has both magnitude and direc- 
tion, and to completely specify a vector one must give both 
its magnitude and direction. 

In everyday life a vector quantity is frequently specified by 
giving its magnitude only. Thus one is satisfied when told that 
the velocity of a train is 60 miles per hour without being told the 
direction in which the train is traveling; the importance of speci- 
fying direction as well as magnitude of vectors is illustrated in 
the following examples: 

A man travels a stretch of 5 miles and then another stretch of 
6 miles; how far from home is he? It is impossible to tell 



THE ARITHMETIC OF PHYSICS. 27 

unless the direction of each stretch is given, or at least the direc- 
tion of one stretch relative to the other. 

A horse pulls with a force of 200 units and another horse pulls 
with a force of 300 units on a post; what is the total force 
exerted on the post? It is impossible to tell unless the direction 
of each pull is given, or at least the direction of one pull relative 
to the other. 

A man walks at a velocity of 4 miles per hour on the deck of 
a boat which is traveling at a velocity of 10 miles per hour; 
what is the total velocity of the man? It is impossible to tell 
unless the direction of the velocity of the man is given relative 
to the velocity of the boat. 

20. Addition of forces. — The following discussion of the addi- 
tion of forces illustrates the method of adding any kind of 
vector quantity. Any number of forces acting together on a body 
are together equivalent to a- single force,* and this single force 
is called the vector sum or resultant of the given forces. 

Addition of two forces. The force parallelogram. — The two 
lines a and h in Fig. 19 represent two forces acting on a body, 
and the line r represents their vector sum or resultant. 

The geometrical relation between a, h and r in Fig. 19 is com- 
pletely shown by the triangle of which the sides are a, h' and r 
(or by the triangle of which the sides h, a' and r). This tri- 





Fig. 19. Fig. 20. 

The force parallelogram. The force triangle. 

angle, which is shown by itself in Fig. 20, is called the addition 
triangle because it determines r as the sum (vector sum) of a 
and h. 

* Except when the forces together constitute a pure turning force or torque. 



■H 



28 



MECHANICS. 



The addition of forces by means of the parallelogram (or tri- 
angle) may be thought of as justified by experiment.* Indeed 
the experimental verification of the addition of forces by means of 
the force parallelogram is usually given as a laboratory exercise 
in elementary physics. Figure 21 shows an arrangement which is 




Fig. 21. 



sometimes used. Two known forces A and B are made to act at 
the point P, and their geometrically constructed resultant C is 
found by the experiment to be equal to the weight W which is 
known. 

Addition of any number of forces. The addition polygon. — 
The lines a, h, c and d in Fig. 22a represent four forces acting on 
a body. To find the vector sum or resultant of these forces start 





^f 



Fig. 22a. Fig. 22b. 

* As a matter of tact, however, the addition of forces by the force parallelo- 
gram is reducible to the addition of displacements. The argument mvolved 
in this reduction constitutes a mathematical proof of the addition of forces by 
the force parallelogram. One form of this mathematical proof is given in Art. 60 
of Franklin and MacNutt's Mechanics and Heat, The Macmillan Co., 1910. 



THE ARITHMETIC OF PHYSICS. 29 

at any point in Fig. 22b and draw a line representing force a; 
from the end of a draw a line representing force b ; from the end 
of b draw a line representing force c; and so on. The point P 
is finally reached, and the line OP represents the desired vector 
sum or resultant. To prove the correctness of this construction 
draw the addition triangle for a and b ; then draw the addition 
triangle for (a + b) and c; then draw the addition triangle for 
{a -\- b -\- c) and d. 

21. Relation among a number of forces whose vector sum or 
resultant is zero. — If in traveling over the lines a, b, c and d in 
Fig. 22b one should come back to the starting point 0, then the 
vector sum or resultant of the forces a, b, c and d would be zero. 
That is, the vector sum or resultant of a number of forces is ?ero 
when the forces are parallel and proportional to the respective 
sides of a closed polygon and in the directions in which the re- 
specttve sides would be passed over in going round the polygon. 
Thus Fig. 23a represents five forces acting on a body, and the 
vector sum or resultant of the five forces is zero as shown in 
Fig. 236. 





Fig. 23a. Fig. 23b. 

When the vector sum of a number of forces is zero, the forces 
together have no tendency to produce translatory motion. 

22. Resolution of a force into parts. The components of a 
force. — Any force is exactly equivalent to and may be replaced 
by a number of forces of which it is the vector sum, or in other 
words a force may be broken up or resolved into parts. Thus the 



30 



MECHANICS. 



vector sum of the two forces X and F in Fig. 246 is the force F 
in Fig. 24a, that is, the two forces X and Y are together exactly 
the same thing as the force F, or, in other words, X and Y are 
two parts into which F can be divided. Now the parts of a 
thing are sometimes called its components, thus the components 
of ordinary concrete are cement and sand and crushed stone; 
therefore the two parts of F in Figs. 24a and 246, namely, X and 
Y are called the components of F\ indeed X and Y are called 
the rectangular components of F because X and Y are at right 
angles'^ to each other. 




Fig. 24a. 



Fig. 24&. 



Fcosf 




Fsin^ 



Fig. 24c. 



Consider the angle 6 in Fig. 24a. 

X is the component in the figure which bounds the angle 0, 
and X is equal to F cos 6. 

Y is the component in the figure which does not bound the 
angle 6, and Y is equal to F sin 6. 

A convenient memory rule covering this matter is to associate 
the in cosine with the in bounds, and to associate the ab- 
sence of in sine with the phrase does not bound. This rule is 
illustrated in a general way in Fig. 24c. 

Example. — Figure 25 shows a force F acting on a car. The 
single force F is exactly equivalent to the two forces X and F. 
The force F has no effectf in helping to move the car, so that 

* Of course a force can be resolved into components which are not at right 
angles to each other, but there is no need to discuss such a case. 

t The force Y has an indirect effect on the motion of the car because it pulls 
the flanges of the wheels against the rail and causes an increased amount of friction. 



THE ARITHMETIC OF PHYSICS. 



31 



the force X is the part of F which is effective in moving the 
car. When the angle 6 is zero, the component X is the same as 



n n 



^wn 



car 




x-axi8 



Fig. 25. 



F and the component Y is zero; the component Y grows 
larger and larger and the component X grows continually smaller 
as d increases from zero to 90°; and when the angle 6 is 90° 
the component X is zero and the component Y is the same 
as F. 

What is stated above concerning the resolution of forces is 
applicable to any vector quantity. Thus we may resolve a 
velocity into components as indicated in Figs. 24a, 24b and 24c. 

23. Constant and variable quantities. — In the study of physics 
we have to do with quantities which do not change (constant 
quantities) and with quantities which do change (variable quan- 
tities) . 

Variable quantities fall into two* important classes, namely, 
quantities which vary from instant to instant in time, and 
quantities which vary from point to point in space. The amount 
of water in a leaking pail is an example of a quantity which 
varies from instant to instant. Atmospheric pressure is an 
example of a quantity which varies from point to point in space 
because the pressure of the air is different at different places. 

In the study of phenomena which depend upon conditions 

This friction is a new force in addition to F; but if F were the only force, then its 
effectiveness in moving the car would be exactly the same as the force X. 

* A third class includes fictitious quantities which are arbitrarily assumed to 
vary. For example, let x and y be two quantities such that y = x^; a funda- 
mental problem in calculus is to determine how fast y changes when x is assumed 
to change at a definite rate. 



32 MECHANICS. 

which vary from Instant to Instant, It Is necessary to direct the 
attention to what takes place at an instant; or, in other words, 
to direct the attention to what takes place during an indefinitely 
short interval of time; or, borrowing a phrase from the photog- 
rapher, to make a snap-shot of the varying conditions. 

In the study of phenomena which depend upon conditions 
which vary from point to point in space, the attention must be 
directed to what takes place at a point, or, in other words, to 
what takes place In an indefinitely small region in the neighbor- 
hood of the point. 

This paying attention to what takes place during an Indefi- 
nitely short interval of time or in a very small region of space 
does not refer to observation, hut to thinking; it is a mathematical 
method and it is called calculus. No one can appreciate the 
science of physics without some understanding of the method 
of calculus, and therefore the essential features of this method 
are set forth in the following articles. 

24. Value at a given instant. Value at a given point. — Let y 

be the amount of water in a leaking pail. Evidently y is 3, 
changing quantity. If the leak were stopped at any given instant 
there would be a definite quantity of water left in the pail, and 
therefore there is a definite quantity of water in the pail at each 
Instant even while the leak continues. 

An Iron rod standing with one end In a fire has a definite tem- 
perature at each point although the temperature varies from 
point to point along the rod. 

What is called a discontinuous variable may not have a definite 
value at each instant or a definite value at each point. This matter 
is discussed in Art. 29. 

25. Rate of change of a quantity which varies from instant 
to instant. — Consider a pail into which water is flowing in a 
small stream. Let y be the amount of water in the pail. Then 
y is evidently a changing quantity. Consider the amount of 
water that flows into the pail during a short Interval of time A/; 
this amount of water is evidently the increment of y during the 



THE ARITHMETIC OF PHYSICS. 33 

short interval of time, and it is to be represented by the symbol 

Ay 
Ay. The quotient — is called the average rate of change cf y 

during the interval A/.* 

If the inflowing stream of water is constant, say, always 2 cubic 

inches of water per second, then if At is chosen smaller and 

smaller, the value of Ay will be proportionately smaller and 

Ay 
smaller, and the value of the quotient —-- will be exactly 2 cubic 

inches per second whatever the duration or length of the time- 
interval At. 

If the inflowing stream is. variable, then the value of the 

A3' 
quotient —7 will not be the same for different values of A/, but 

the amount of water which flows into the pail during a very short 
interval of time will be very nearly proportional to the interval. 
For example, a certain amount of water W flows into the pail 
during a particular gne- thousandth of a second. Imagine this 
particular one-thousandth of a second to be divided in halves; 
then only a very little more than hW will flow into the pail 
during one of the half-thousandths of a second and a very little 
less than \W will flow into the pail during the other half- 
thousandth of a second. // a shorter and shorter interval of time 
he taken the amount of inflow of water will be more and more 
nearly in EXACT proportion to the duration of the interval. 

A3' 
This means that the quotient — approaches a definite limiting 

value as At and A3' both approach zero ; and this limiting value 

A3' . 
of — - is called the rate of change of y at a certain instant or 

the instantaneous rate of change of y. 

A3' . dy 

The limiting value of — - is always represented by — which 

i\L ai 

* Thus if y is the amount of money a man has saved, then, if he saves $20 
more during 30 days, we have Ay = I20 and At = 30 days, and the average rate 
of saving during the 30 days will be 66| cents per day. 
4 



34 MECHANICS. 

is a single symbol and it means the rate of change of y at a given 
instant. 

Nearly everyone falls into the idea that such an expression as 
10 feet per second means lo feet of actual movement in an actual 
second of time, but a body moving at a velocity of lo feet per 
second might not continue to move for a whole second or its 
velocity might change before a whole second has clasped. Three 
cubic inches per second is the same rate of inflow of water into a 
pail as 2,070 cubic yards per year, but to specify the rate of inflow 
in cubic yards per year does not mean that a whole cubic yard of 
water flows into the pail nor does it mean that the inflow continues 
for a year. A man does not need to work for a whole month to 
earn money at the rate of $60.00 per month, nor for a whole day 
to earn money at the rate of $2.00 per day. A falling body has 
a velocity of 19,130,000 miles per century after it has been falling 
for one second, but to specify its velocity in miles per century 
does not mean that it falls as far as a mile nor that it continues to 
fall for a century! The units of length ajid time which appear 
in the specification of a velocity are completely swallowed up, 
as it were, in the idea of velocity; and the same thing is true of 
the specification of any rate. 

26. Example showing determination of an instantaneous rate 
of change. — Let us assume that the amount of water in the pail 
in the discussion of Art. 25 is proportional to the square of the 
time / which has elapsed since a chosen instant. Then we may 

write 

y = kf (i) 

where ^ is a constant. A moment later t has increased and of 
course y has increased also. Let us represent the new value of t 
by t -\- M and the new value o{ y hy y -\- Ay. Then, since 
equation (i) is assumed to be true for all values of t and y, we 
have 

y + Ay = k{t-\- Aty 
or 

y -{- Ay = kf + 2kt'At + k{Aty (ii) 



THE ARITHMETIC OF PHYSICS. 35 

Subtracting equation (i) from equation (ii) member by member, 

we have: 

^y = 2kt'M + k{MY (iii) 

Whence, dividing both members by A/, we have: 

~ = 2kt^k'M (iv) 

Now it is evident that k • At approaches zero as At approaches 

Ay 
zero, and therefore the Hmiting value —-7 (as At approaches 

dy , , . Ay 

zero) is 2kt. Hence, writing -r- for the limiting value of — , 

we have: 

That is, the rate of change of y is at each instant equal to 2kt. 

Observation or thinking; which? — The discussion in Art. 25 
presents a serious difficulty. How is one to know the increase 
of the amount of water in the pail during an indefinitely short 
interval of time? One certainly cannot measure it. Observa- 
tion is entirely out of the question in the consideration of in- 
definitely small changes which occur during indefinitely short 
intervals of time. One cannot observe such things, one can only 
think about them. This same remark applies to the discussion 
of gradient in the following two articles. 

27. Gradient of a quantity which varies in space. — Consider 
an iron rod which stands with one end in a fire. Then the tem- 
perature of the iron varies from point to point along the rod. 
Consider two points very near together, let AT" be their difference 
in temperature and let Ax be their distance apart. The quotient 

AT 

— - approaches a definite limiting value when Ax is chosen 

AT . 
smaller and smaller, and this limiting value of ■-— is called the 

Ax 

temperature gradient at the point (degrees per centimeter or per 
inch) . 



36 MECHANICS. 

28. Example showing the determination of a gradient at a 
point. — Consider a metal bar AB, Fig. 26. "Suppose the tem- 

fOld T hof 

A i-^ J^ ^J^~ ^^^ "^ ^ ^ 

Fig. 26. 

perature of the bar to be zero at the end A , and let us assume the 
temperature T at the point p to be: 

T = kx^ (i) 

where x is the distance from A to p, and ^ is a constant. 
Under these conditions let it be required to find the temperature 
gradient at the point p. Now equation (i) is true for every value 
of X and T. Therefore writing x -\- Ax for x and writing 
T + AT for T we have 

T -\- AT = k{x -\- AxY (ii) 

or 

T + AT = kx'^ + 2kx'Ax + k{Axy (iii) 

Subtracting equation (i) from equation (iii) member by member, 

we have: 

AT = 2^x-Ax + k{Axy (iv) 

whence 

AT ^ ^ , , 

-~ = 2kx -\- k'Ax \y) 

Ax 

Now Iz-Ax approaches zero as Ax approaches zero, and it is 

AT . 
therefore evident that the limiting value of —— is 2kx. There- 

Ax 

fore, writmg — for the limitmg value of - — , we have: 

T- = 2kx (vi) 

ax 

It is evident from this equation that the temperature gradient is 
zero at the end A of the bar where x = o; and it is evident that 
the temperature gradient grows greater and greater (steeper and 



THE ARITHMETIC OF PHYSICS. 37 

steeper) at points farther and farther from A where x is larger 
and larger. 

29. Continuous variables and discontinuous variables. — A 

quantity which changes by sudden jumps is called a discontinuous 
variable. For example the amount of money one has is a dis- 
continuous variable, because a debt is made on the instant that 
one decides to accept a purchase ; that is to say, money is spent in 
lumps, and a lump of money is spent during an indefinitely short 
time. The amount of money spent during an interval of time does 
NOT become more and more nearly proportional to the interval cCs the 
interval grows shorter and shorter, and consequently it is meaning- 
less to speak of the rate of spending money at a given instant. If 
3' is a discontinuous variable and if the time-interval A^ happens 
to include a jump in the value of y, then the value of A3; remains 

Ay 
finite as At approaches zero and the quotient — approaches in- 
finity. The rate of change of a discontinuous variable at a given 
instant is unthinkable. 

The amount of water in a pail as considered in Art. 25, and the 
temperature along the iron rod as considered in Art. 28 are 
examples of continuous variables. 

30. Velocity and its variation in time.* — Consider a car moving 
along a straight track and covering equal 

distances in equal times. Under these condi- f yv 

tions the velocity of the car is constant and / / / 

equal to /// where / is the distance traveled / y^' j 

by the car in t seconds. // / 

Let the arrow v in Fig. 27 represent the 0^-— ^ ^' 

velocity of a body at a given instant, and ^ig. 27. 

let the arrow v' represent the changed 

velocity of the body bd seconds later. Now the arrow Ay 

* The discussion in Arts. 24-28 applies to scalar quantities, whereas velocity 
is a vector quantity. 

The variation of vectors in space is discussed in Franklin, MacNutt and Charles' 
Calculus, Chapter IX; published by the authors, South Bethlehem, Pa., ipis* 



38 MECHANICS. 

represents the velocity which must be added to v to give v', 
and therefore the arrow Av represents the change of velocity of 
the body during the time A^, and the quotient Av/At is the 
average rate of change of the velocity (the average acceleration of 
the body) during the time At. If the time interval At is chosen 
smaller and smaller the quotient AvjAt approaches a definite 
limiting value which is called the instantaneous rate of change 
of the velocity of the body {instantaneous acceleration of the body). 

31. Velocity and acceleration of a body which moves at uni- 
form speed in a circular orbit. — ^A stone of mass m is fixed to a 
string and twirled in a circular path or orbit of radius r. The 
stone always presents the same face towards the center of its 
circular orbit and therefore the stone makes one revolution about 
an axis perpendicular to the plane of the orbit every time it 
moves round the orbit hut we are not here concerned with this 
constant rotatory motion of the stone, we are concerned only with 
its translatory motion as its center of mass describes a circular 
path of radius r, and we may think of the stone as concentrated 
at its center of mass as explained in Art. 9. 

Let n be the number of times the stone moves round its orbit 
per second. Then 2Trrn is the distance traveled by the stone per 
second. That is 2Trrn is the velocity v of the stone. Therefore: 

• V = 27rrn (i) 

In considering the velocity of the stone, however, we must 
always think of its direction as well as its value, and this equation 
gives its value only. Indeed the stone is always traveling in a 
direction at right angles to the string and of course this direction 
is changing continuously. 

To determine the acceleration of the stone let us consider the 
change of its velocity during a very short interval of time, (%t, 
while the stone is moving from P to Q in Fig. 28. The velocity 
of the stone when it is at P is represented by the arrow Vi and 
the velocity of the stone when it is at Q is represented by the 
arrow V2 ; and these two arrows are drawn from a common origin 



THE ARITHMETIC OF PHYSICS. 



39 



0' in Fig. 29. Therefore the short arrow Ap in Fig. 29 repre- 
sents the change of velocity of the stone during the time At. 
Now the arrows Vi and V2 are perpendicular to OP and OQ, 




ow- 



Ad) 



Vi 



^p' 



I 
I 

fAv 



-t. 



Fig. 28. 



Fig. 29. 



respectively, and therefore the angles Acj) in Figs. 28 and 29 are 
the same; also the two velocities Vi and V2 have the same value 
(= 2Trn = v) so that the two arrows Vi and v^ are of the 
same length. Furthermore, if the time- interval M is indefinitely 
short the arc PQ in Fig. 28 is sensib'y a straight line, and the 
length of PQ is equal to v • A/, the distance traveled by the stone 
in the time M. The two triangles OPQ and O'P'Q' are similar 
and therefore we have 



r 



Av 

V 



(ii) 



where v is written for the common value of Vi and v^. There- 
fore, solving equation (ii) for ^ we have: 



X 






(iii) 



and this is indeed the limiting value of AvjAt because we have 
treated the arc PQ as an indefinitely short straight line; but 
the limiting value of AvjAt is the desired instantaneous accelera- 
tion a of the stone when it is at P (the line PQ is indefinitely 



40 MECHANICS. 

short so that the points P and Q are coincident at the Hmit). 
Therefore we have 

a = - (8) 

r 

and it is evident that a is parallel to the radius PO, for the 
velocity gained during a very short time, as represented by At; 
in Fig. 29, is parallel to PO. The acceleration a can be ex- 
pressed in terms of r and n by substituting the value of v 
from equation (i) in equation (8), which gives: 

a =-^TrVr (9) 

Therefore, a body traveling n times per second around a circular 
orbit of radius r gains velocity continuously towards the center of 
the circular orbit at a rate which is equal to ^ir'^nH, or equal to 
v'^lr, where v is the value of the velocity of the body. 

Force required to constrain a body to a circular orbit. — When 
a stone is tied to a string and twirled in a circular orbit, the 
pull of the string on the stone is an unbalanced force because it 
is the only force acting on the stone (the gravity pull of the earth 
on the stone and the f rictional resistance of the air being ignored) , 
and it is this unbalanced pull of the string that produces the 
acceleration as given by equations (8) and (9). Therefore, 
substituting the value of a from equation (8) or (9) in equation 

(i) of Art. 6, we have: 

^2 

F = m- (10) 

r 

or 

F = 4.TrVrm (11) 

where F is the pull of the string in dynes (or poundals), m is 
the mass of the stone in grams (or pounds), r is the radius of 
the circular orbit in centimeters (or feet), n is the number of 
times per second that the stone passes round its circular orbit, 
and V is the velocity of the stone in its orbit ( = 2Trrn) in centi- 
meters per second (or feet per second). 



THE ARITHMETIC OF PHYSICS. 



41 



32. Example of translatory motion in a circle. Locomotive 
on a railway curve. — A locomotive traveling around a railway 
curve is a combination of translatory motion and rotatory 
motion. The locomotive not only moves forwards but it rotates 
about a vertical axis. The rotatory motion can be most easily 
understood if we assume the curve to be a complete circle. 
Then the locomotive will turn once round every time it com- 
pletes the circular path or orbit, and if the locomotive travels at 
constant speed its rotatory motion will he constant in value and 
about a fixed-direction axis. 

The translatory velocity of the locomotive is changing direc- 
tion continuously like a stone twirled on a string, and therefore 
the locomotive has an acceleration equal to v'^jr continuously 
towards the center of curvature of the track, where v is the 
velocity of the locomotive in centimeters per second (or feet per 
second) and r is the radius of the curve in centimeters (or feet) , 
according to equation (8) of Art. 31. 

The earth pulls downwards on the locomotive with a force of 
mg poundals, where m is the mass of the locomotive in pounds 
and g is the acceleration of gravity, according to equation (4) 

A' "^ 

total force acting 

on- locomotive 




Fig. 30. 

of Art. 6. Therefore the track must push vertically upwards 
on the locomotive with a force of mg poundals (as indicated in 
Fig- 30) to support the locomotive. Also the track must push 
on the locomotive with a force of mv^/r poundals, according to 
equation (10), in order to produce the acceleration v'^/r, and this 
force, as shown in Fig. 30, is horizontal and towards the center of 



42 MECHANICS. 

the railway curve. The total force exerted on the locomotive 
by the track is the resultant of mg and mv^/r as shown in Fig. 30. 
It is desirable that the face of the track ff be at right angles to the 
total force with which the track acts on the locomotive, therefore the 
angle of elevation of the track, the angle e, should be equal to 
the angle B. But the tangent of 6 is mv^jr divided by mg, or 
vy{rg). Therefore the angle of elevation of the track should be 
such that 

tan e = — (12) 

rg 

In fact the outer rail is always elevated on a railway curve, 
but of course trains go round a curve at widely different speeds. 
If the velocity of a locomotive is greater than '\rg • tan e [accord- 
ing to equation (12)], the flanges of the locomotive wheels press 
against the outer rail. If the velocity of a locomotive is less 
than Vrg • tan e, the flanges press against the inner rail of a 
curve. 

Note.- — The above discussion refers to a lone locomotive. In 
the case of a car in a long train the conditions are different be- 
cause a train is under tension like a belt. 

When a locomotive is on a straight track it does not rotate, and after the 
locomotive has entered a curve it does rotate about a vertical axis as above ex- 
plained. Therefore as the locomotive enters a curve this rotatory motion must be 




Fig. 31a. 

suddenly established, and it is established by an excessive side-force exerted against 
the flanges of the front wheels of the locomotive by the outer rail. From the 
point of view of a man on the locomotive the flanges of the front wheels of the loco- 
motive push with excessive force against the outer rail as the locomotive enters a 
curve. This action is called nosing. It is especially troublesome in the case of a 
locomotive with a short wheel base because it is the torque action of the side force 



THE ARITHMETIC OF PHYSICS. 43 

on the front wheels that counts, and for a given side force this torque action is 
proportional to the length of wheel base. The electric locomotives of the New York, 
New Haven and Hartford Railroad, as first constructed, had very short wheel 
bases, and the nosing was troublesome until pilot trucks were placed at the ends. 
Figure 31a shows a straight portion of track changing abruptly to a circular 
curve at the point a, and Fig, 316 shows the same straight portion changing 



Fig. 31&. 

gradually into the same circular curve. The slow transition in Fig. 31& constitutes 
what is called an easement curve, and its object is to avoid the effects of abrupt entry 
of a locomotive into a curved portion of track as above described. The locomotive 
is set in rotation gradually as it traverses the easement curve. 

33. Example of translatory motion in a circle. The cen- 
trifugal drier. — The centrifugal drier which is used in laundries 
and sugar refineries is a rapidly rotating bowl, with perforated 
sides, into which the material to be dried is placed. The action 
of the centrifugal drier may be explained by considering the mo- 
tion of a very small portion, ab, Fig. 32, of the material to be 
dried. Such a small portion travels in 
a circular path as the bowl rotates, as 
indicated by the dotted circle. 

The two solid particles a and b rest / 
directly or indirectly against the side of \ 
the bowl, and the force F which must be \ 

exerted on the drop of water d to keep it ^-^^ ^^ 

in its circular path is exerted on the drop ^^ 

Fig. 32. 

by the two particles. The drop is said to 

adhere to a and b. But the force which can be exerted on the 
drop by its adherence to a and b is limited, and, therefore, 
when the bowl rotates at high speed, the particles a and b 
cannot exert on J a force sufficient to keep d in its circular path. 



/ 
/ 

I 




44 



MECHANICS. 



Consequently d separates from a and h and moves off along 
the tangent /. Imagine a piece of wet cloth jerked to one side 
very quickly. If the cloth does not exert sufficient force on the 
water to give it the same acceleration as the jerked piece of cloth 
most of the water will be left behind ! This action is essentially 
the same as the action of the centrifugal drier. 

34. Example of uniformly accelerated translatory motion and 
rotatory motion. — An interesting example of uniformly acceler- 
ated motion which involves both translatory motion and rotatory 
motion is as follows. A wheel and axle is mounted on ball 



axle 



wheel 




axle 



G 



string 



? 



m 



Fig. 33- 



bearings so as to be as nearly as possible frictionless, and the 
wheel and axle is set rotating by a falling weight m as indicated 
In Fig. 33. 

Let ^ be the spin velocity of the wheel and axle and v the 
downward velocity of the weight. Then 



V = rs 



(i) 



where r is the radius of the axle. This equation is derived as 
follows: If 7t were the speed of wheel and axle in revolutions 
per second, then n laps or turns of cord would be unrolled from 



THE ARITHMETIC OF PHYSICS. 45 

the axle per second; but the length of each lap or turn is 27rr, 
so that the length of cord unrolled per second would be 2Trrn 
which is the velocity of the weight. Now n is the speed of the 
wheel and axle in revolutions per second and 2'Kn is its speed 
in radians per second {= s). Therefore v = rs. 
From equation (i) we get* 

a = ra (ii) 

where a is the rate at which v is changing (the acceleration of 
the weight m), and a is the rate at which ^ is changing (the 
spin-acceleration of wheel and axle). 

The force F with which the earth pulls on the body m is 
mg poundals where m is the mass of the body in pounds, and 
g is the acceleration of gravity in feet per second per second. 

A portion of the force mg is used to accelerate the body w, 
and the portion so used is ma poundals. The remainder, 
{mg — ma) poundals, is the force G which is producing the spin- 
acceleration a of the wheel and axle. The torque action of this 
force about the axis of the wheel is {mg — ma)r, and this 
torque is equal to Ka according to equation (7) of Art. 14. 
That is 

Ka = {mg — ma)r (iii) 

where K is the spin-inertia of the wheel and axle about the axis 
of rotation. Substituting the value of a from equation (ii) in 
equation (iii) and solving for a, we get 



/ "" \ r \ 




* The arithmetical argument by means of which equation (ii) is derived from 
equation (i) is called differentiation. In this particular case the matter is extremely 
simple: If v is always r times as large as 5 it must always change r times 
as fast as s, r being a constant, but the rate of change of 5 is the spin-acceler- 
ation a, and the rate of change of v is the translatoiy acceleration a. There- 
fore a = ra. Reduced to homely terms this proposition is as follows: If James 
always has 10 times as much money as John, then James must always gain or lose 
money 10 times as fast as John. 



46 



MECHANICS. 



Therefore, if K,m, r and g are known, we can calculate the 
value of a. 

The arrangement shown In Fig. 33 is frequently used in the 
laboratory for the determination of the spin-inertia of a wheel 
as follows: The acceleration a of the body m is constant, and 
therefore if the weight starts from rest it will fall a distance d 
in / seconds, where 

(v) 



d = \af 



as explained in Art. 16. Substituting the value of a from this 
equation in equation (iv) and solving for K we get 

fg 



K 



mr 



2d 



— I 



) 



(vi) 



in which m is the known mass in pounds of the falling weight 
in Fig. 33, g is the acceleration of gravity (supposed to be 
known), d is the observed distance fallen by the body in t 
seconds, and r is the measured radius of the axle in Fig. 33. 

35. The gyroscope.— Figure 34 
shows a rapidly spinning wheel 
mounted in a ring which rests on 
a post BO. The pull of gravity 
on the ring and wheel produces 
an unbalanced torque about the 
axis 0*, and this unbalanced 

'l^ull of gravity *°^^"^ ^^"^^^ ^^^ ^^^^ ^^^ ^'']^ 
Oa to swing round and round in 

a horizontal plane (end a of 
axle moves away from reader). 
It is to be noted that the axis of the torque due to gravity, the 
axis 0, remains at right angles to the plane BOa as the axle and 
ring swing round. A wheel and axle mounted in this way consti- 
tute what is called a gyroscope. 

The action of the gyroscope as here described is an example 
of rotatory motion which is exactly analogous to the translatory 

* Horizontal axis at right angles to the plane of the paper in Fig. 34. 




nn/t>) 



table top 

Fig. 34- 



THE ARITHMETIC OF PHYSICS. 



47 



motion of a stone in a circular orbit, and the best elementary 
explanation of the action of the gyroscope may be reached by 
developing this analogy as follows: 

Translatory motion in a circle. — Translatory motion carries a 
body from one place to another place, whereas rotatory motion 
does not. Therefore to appreciate the very close analogy 
between gyroscopic motion and the translatory motion of a 
stone in a circle one should think only of the varying velocity of 
the stone without thinking of its changing position. 

Figure 35 shows a stone which is being twirled on a string, 
and Fig. 36 shows an arrow v drawn from a fixed origin 0' so 
as to represent the velocity of the stone, and an arrow F which 
represents the force with which the string pulls on the stone. 
Evidently the two arrows sweep round the point 0', in the 



/orbit 



\ 







string 



I 
I 
» 
/ 

/orbit 



-L. 



/ 



/ 



stone 

Fig. 35- 




Fig. 36. 



direction of the curved arrow 12 as the stone moves round its 

circular orbit, but F and v remain at right angles to each other. 

During a very short interval of time At, the force F produces 

F 
an amount of velocity Av in its direction, and Av = ~ -At, 

tn 

according to equation (i) of Art. 6. This is evident when we 

consider that Fjm is the rate of gain of velocity by the stone 

(acceleration of stone) and that acceleration multiplied by time 

gives gained velocity. 

Now the angle A0 in Fig. 36 is very small so that we may 



48 



MECHANICS. 



think of the very short arrow Az; as a circular arc, and therefore 
the angle A0 expressed In radians Is equal to hviv (arc divided 
by radius) . Consequently the radians per second turned by the 
arrows v and F is: 

A</) Av 

At V 

or, substituting — -A^ for Az^, we have 
m 

F 
n = — 

mv 



or 



F = mvQ, 



(i) 



That is to say, the force F (acting always at right angles to v 
as shown In Fig. 36) required to make the velocity of the stone 
(the arrow v In Fig. 36) sweep round at a speed of fi radians per 
second as Indicated In Fig. 36 Is equal to mvQ., where m Is the 
mass of the stone and v Its velocity as It travels In its circular 
orbit. Equation (I) can be easily reduced to the form of equa- 
tion (10) or equation (11) of Art. 31. 

Gyroscopic motion. — Figure 37 represents a gyroscope as seen 
from above, and Fig. 38 shows an arrow 5 drawn from a fixed 




top view 

Fig. 37- 




Fig. 38. 



origin 0' so as to represent the spin- velocity of the wheel, and an 
arrow T which represents the torque action of gravity on the 
ring and wheel. The two arrows 5 and T sweep round the 
point 0' In the direction of the curved arrow fi as the gyroscope 
moves, but ^ and T remain at right angles to each other. 



THE ARITHMETIC OF PHYSICS. 49 

During a very short interval of time A/, the torque T pro- 
duces an amount of spin-velocity A^ about its axis, and 

T 
As = -^'At, according to equation (7) of Art. 14. This is 
K. 

evident when we consider that T/K is the rate of gain of spin- 
velocity by the wheel (spin-acceleration of wheel) and that 
spin acceleration multiplied by time gives spin- velocity gained. 
Now the angle Acf) in Fig. 38 is very small so that we may 
think of the very short arrow As as a circular arc, and therefore 
the angle A0 expressed in radians is equal to As Is (arc divided 
by radius). Consequently the radians per second turned by the 
arrows s and T is: 

A(t) As 
At s 

. . T 

or, substituting t^-A/ for A^, we have: 
A. 

T 



or 



T = Ks2 (ii) 



That is to say, the torque T (acting always at right angles to s 
as shown in Fig. 38) required to make the axis of the spinning 
wheel (the arrow 5 in Fig. 38) sweep round at a speed of 
radians per second as indicated in Fig. 38 is equal to KsQ, where 
K is the spin-inertia of the wheel about its axis of spin and s 
is the spin velocity of the wheel. 

Gyrostatic action reduced to simplest terms. — Figure 39a 
represents a bicycle wheel not spinning, but supported at one 
end of its axle at as shown. The pull of gravity on the 
wheel constitutes a turning force or torque about an axis at 
perpendicular to the plane of the paper; and after this torque 
has acted on the wheel for a short time every particle in the upper 
half of the wheel has an eastward velocity as indicated by the arrow A 
and every particle in the lower half of the wheel has a westward 
5 



50 



MECHANICS. 



velocity as indicated by the arrow B. This is the effect of the 
gravity torque on the wheel when it is not spinning. 

Let us now suppose that the wheel in Fig. 39a is spinning as 



west O 




north 



north 



O 



east 



west 



east 



west 




south 
top view 

Fig. 39b. 



south 
top view 

Fig. 39c. 



indicated in Figs. 39^ and 39c, both of which are top views. 
The gravity torque acting for a short time about the axis 00 
turns the axis of spin ^ from the position shown in Fig. 396 to 
the position shown in Fig. 39c; every particle in the upper half of 
the wheel in Fig. jgc has an eastward velocity-component as 
indicated by the arrow A in Fig. jqc, and every particle in the lower 
half of the wheel has a ivestward velocity-component. 

Therefore, eastward and westward velocities are produced by 
the gravity torque in Figs. 39^ and 39c when the wheel is spinning 
just as eastward and westward velocities are produced in Fig. 39a 
when the wheel is not spinning. The precise mathematical 
formulation of the production of eastward and westward veloci- 
ties in Figs. 39& and 39c constitutes the fundamental theory of 
gyroscopic action. 

36. Uses of the gyroscope.* — The simplest use of the gyro- 
scope is for automatic rudder control in a torpedo. A gyroscope 
is mounted in the torpedo with its axis free so that when the 

* The use of the gyroscope in the torpedo, the use of the gyroscope for steadying 
a vessel at sea, and the use of the gyroscope for equilibrating a monorail car are 
described on pages 66 to 78 of H. Crabtree's Spinning Tops and Gyroscopic Motion, 
Longmans, Green & Co., London. 



THE ARITHMETIC OF PHYSICS. 51 

torpedo turns to one side or the other the gyroscope axis retains 
its direction unchanged, and the relative motion of gyroscope 
axis and torpedo frame is utihzed to open valves which admit 
compressed air to a cy Under and piston arrangement, thus moving 
the piston and the rudder to which the piston is connected. 

The gyroscope may be used in the same way to give automatic 
control of the rudders of an aeroplane. 

The gyro-compass. — It has been known for many years that 
a gyroscope mounted with its axis in a horizontal plane and 
free to turn in that plane (about a vertical axis) would oscillate 
back and forth through the north-south position and finally 
come to rest with its axis pointing due north and south. The 
modern gyro-compass depends on this effect, and the design of a 
practicable gyro-compass involved two distinct problems, namely, 
{a) to support the gyroscope with extremely small "pivot" 
friction and at the same time maintain the spinning motion of the 
gyroscope wheel, and (6) to eliminate the above described oscil- 
latory motion and keep the gyroscope axis pointing steadily 
north and south. The necessity of this second condition will 
be understood from the fact that the period of the oscillations 
of a gyro-compass is five or six hours! Imagine an ordinary 
compass needle oscillating back and forth through an unknown 
range or amplitude once every five or six hours; it would require 
a careful all-day study of such a needle to know the direction 
in which it would point if it were standing still! 

Given a rotating wheel and axle. Looked at from one end the 
motion appears to be counter-clock-wise, let this be called the 
positive end; looked at from the other end the motion appears to 
be clock-wise, let this be called the negative end. Following is 
an explanation of the tendency for the positive end of a properly 
mounted gyroscope wheel and axle to point north. 

Figure 40 shows a spinning gyroscope wheel mounted in a 
frame and hung on a pivot, and the positive end of the gyro is 
shown pointing east with the frame balanced in a horizontal 
position. The rotation of the earth an its axis soon brings the 



52 



MECHANICS. 



piumb line into a position PP, or in other words, the gyroscope 
frame soon comes into an inclined position as shown in Fig. 41 
so that the gravity pull of the earth on the frame and wheel 



upi 

pI 

I 



frame 



west 



axle ! T 



fc 



''pivot 



wheel 



east 



4-^ 




e€ist 



I down 

Fig. 40. 



exerts on the frame a torque UD which tends to bring the 
suspended frame into a horizontal position. The effect of this 
torque, however, is shown in Figs. 42 and 43. Figure 42 is a top 
view of Fig. 41, the frame being omitted for the sake of clearness. 
The inclination of the frame in Fig. 41 is greatly exaggerated, 
indeed this inclination is hardly perceptible as seen from above 
in Fig. 42. The torque UD is represented by the arrow T 
in Fig. 43, and the spin which is produced by T during a very 
short time is represented by As. When As is added to the 



north 



west 



U 



I 



D 

^east 



west 



north 




\ 
\ 



east 



south 

Fig. 42. 



south 

Fig. 43. 



existing spin it gives the resultant spin s' and the axis of the 
wheel will have turned from 5 to s'. This shows that the 
positive end of the gyro axis is turning towards the north. 



THE ARITHMETIC OF PHYSICS. 



53 



-jc-^ 



Exactly the same kind of argument will show that the positive 
end of the gyro axis turns towards the north when it is pointing 
westwards; and a slightly modified argument shows that the 
positive end of the axis turns towards the north in whatever 
direction (other than north) it may be pointing.* 

The above described action is exactly what takes place at the equator. At any- 
place north or south of the equator the spin-motion of the earth has two components, 
namely, (a) a component about a vertical axis, and (b) a component about a 
horizontal north-south axis. The latter component acts on the gyroscope as 
represented in Figs. 40 to 43 and causes the positive end of the gyroscope axis to 
turn towards the north; and the former component leads to a slight upward tilt of 
the north end of the axis of the gyro-wheel (as it stands in a north-south position), 
and the torque action exerted on the frame by gravity because of this upward 
tilt causes the axis of the gyroscope to follow the true north-south position in spite 
of the rotation of the horizon about a vertical axis, the component a above 
mentioned. 

37. Hookers law of elasticity f and simple harmonic motion. — 

Figure 44 shows a heavy ball B fixed to one end of a flat spring, 
the other end of the spring being clamped in a 
vise. The figure represents the ball B at rest 
and the spring straight (unbent) . This position 
of the ball is called its equilibrium position. If 
the ball is pushed sidewise the spring is bent, 
and the force required to bend the spring is pro- 
portional to the sidewise movement, x, of 
the ball, or equal to kx, where ^ is a constant 
for the given spring; this is true if the sidewise 
movement is not too great. This proportional 
relationship between a distorting force and 
the distortion which it produces in an elastic 
body like a spring was discovered by Robert 
Hooke in 1676 and it is called Hooke's law. 

Imagine the ball in Fig. 44 to be held steadily in the dotted 

* There is an interesting reaction which tilts the frame about an east-west 
axis when the positive end of the gyro axis is pointing other than due east or west 
and while the positive end of the gyro is turning towards the north. 

t A fairly complete elementary discussion of the theory of elasticity is given in 
Chapter VII of Franklin and MacNutt's Mechanics and Heat, The Macmillan Co.„ 
New York, 1910. 



ball BiM 



flat spring 







I! 

i! 



II 
If 

11 



Fig. 44. 



54 



MECHANICS. 



position ; then an outside force hx must be exerted on the ball, 
and the spring exerts an equal and opposite force on the hall; there- 
fore we may write 

F = - kx (13) 

where F is the force exerted on the ball by the spring, ^ is a 
constant for the given spring, and x is the distance shown in 
Fig. 44. The negative sign indicates that F is towards the 
left when x. is towards the right, and that F Is towards the right 
when X is towards the left. 

If the ball in Fig. 44 is pushed to one side and released it will 
vibrate back and forth; this vibratory motion of the ball is of a 
kind that is very important in the theory of sound, and it is 
called simple harmonic motion. In the following discussion 
simple harmonic motion is defined ideally, and the force action 
which is required to maintain it is determined theoretically. 
The force action thus determined is exactly in accordance with 
equation (13), and from this we infer that the ball B in Fig. 
44 does, in fact, perform simple harmonic motion when it vibrates. 

Definitions. — Simple harmonic motion is the projection on a 
fixed straight line of a point which moves round and round a 
circle at constant speed. Consider, for example, the point P', 





Fig. 45- 



Fig. 46. 



Fig. 45, which moves round and round the circle at constant 
speed, making n revolutions per second. Then the point P 
(the projection of P' on the fixed straight line AB) vibrates 
back and forth along AB; making n "round-trip" excursions 
per second. One "round-trip" excursion is called a cycle, the 
number of cycles per second is called the frequency of the vibra- 



THE ARITHMETIC OF PHYSICS. 



55 



tions, and the distance OA or OB is called the amplitude of 
the vibrations. 

Figure 46 shows two points P' and Q' moving at the same 
speed around a circle, and P and Q are two points performing 
simple harmonic motion of the same frequency. The vibrations 
of P and Q are said to differ in phase and the angle d is called 
their phase difference. 

Acceleration of and force action upon a body which performs 
simple harmonic motion of frequency n cycles per second. — 
The point P** Fig. 47, makes V 
revolutions per second round the 
circle, and its projection on AB, 
that is the point P, makes -^ 
" round-trip " excursions (cycles) 
along AB per second. 

The distance x (namely, OM is the 
horizontal projection (horizontal 
component) of the line 0P\\ the 
velocity of pq is the horizontal 
component 01 the velocity of P ; 
and the acceleration of M is 
the horizontal component of the acceleration of P\. But the 
acceleration of P\ is represented by the arrow a' and it is equal 
to /^irVr according to equation (9) of Art. 31 ; and the accelera- 
tion of J^is represented by the arrow a. Therefore from the 
similar trl'angles jb'P^jC ^nd OP^W we have: 
/ ' '-IT// 

^- 




Fig. 47- 



'h 



where x is written for 



# 



TW 



or 



\ 



X 



a = 




Hence, using the value 



f for <z', we have: 



a = — /^ttVx 



oQ^ ^^ 






^ 



(14) 



> 



r 



56 MECHANICS. 

The negative sign is used because a is to the left when x is 
to the right, and vice versa. 

The acceleration of the point P being known it is possible 
to find an expression for the force which would have to act on a 
body of mass m moving back and forth like the point P. It is 
only necessary to substitute the value of a from equation (14) 
in equation (i) of Art. 6 which gives: 

F = — /i^TrVx-m 
or 

F— — {/\TrVm)'X (15) 

Now the factor in the parenthesis is a constant for a given 
body (given m) vibrating at a given frequency (given n). 
Therefore let us represent this factor by a single letter k so that 

k = /^irVm (16) 

Then equation (15) becomes: 

F = - kx (17) 

which is identical to equation (13). Therefore the body B in 
Fig. 44 performs simple harmonic motion when it is set vibrating. 

Numerical example. — Suppose it is found by trial that a 
force of 250 poundals is required to hold the body B in Fig. 44 
at a distance of 0.0625 foot to one side of its equilibrium position. 
Then using these values for F and x in equation (13) we get 
k = 4,000 poundals per foot. Suppose also that the body B 
in Fig. 44 has a mass of 10 pounds {— m). Then substituting 
4,000 poundals per foot for k in equation (16) and substituting 
10 pounds for m we can calculate the value of n, and it comes 
out to be w = 3.12 cycles per second. 

In this example nothing is said as to the amplitude of the 
vibrations of Bj and indeed the frequency is the same whatever 
the amplitude may be. 

38. The ideal simple pendulum. — The ideal simple pendulum 
consists of a small ball B suspended by a string or thread of 



THE ARITHMETIC OF PHYSICS. 



57 



negligible mass as shown in Fig. 48. Let / be the distance OB, 
and let g be the acceleration of gravity. Then the number of 
round trip vibrations (cycles) per second of the pendulum when 
it is set vibrating is 



n 



= h\ 



(18) 



(point of suspension) 



provided the pendulum swings through a small amplitude, this 
equation is not true if the amplitude 
is large. 

Proof. — Let m be the mass of 
B. Consider the state of things when 
B is at Q. The pull of gravity on B 
is a vertical force of mg poundals,* 
and this force may be resolved into the 
two components F and G. The 
force G is balanced by the pull of 
the thread, and the force F is un- 
balanced. From the geometry of the 
diagram we have: 



F = mg sin 



(i) 




but if (f) is moderately small the value 
of <f> in radians is very nearly equal to sin 0. Therefore, ap- 
proximately, we have 

F = mg(j) 

which may be written thus : 



F = f -Z^ 



or, since /0 = x, we have: 






(ii) 



mg 



Now the factor -y is a constant, and it may be represented 



* Of course c.g.s. units can be used. See Art. 18. 



58 




MECHANICS. 


by 


a single letter 


kj where 








k -- 


mg 
I 


so 


that equation (ii) becomes 








F 


= kx 



(iii) 



(iv) 

Of course a negative sign should be used in all these equations 
because F is to the left when x is to the right. Therefore 
equations (iii) and (iv) are identical with equations (i6) and (17). 
That is, the factor k in equation (iii) is equal to ^irVm accord- 
ing to equation (16). Therefore from equation (iii) we have: 

mg 



or 

n 



= -^ J (18) 

27r \/ 



Numerical example. — A simple pendulum for which / = 94.5 
centimeters is observed to make 100 vibrations (cycles) in 1 95. 1 
seconds at Bethlehem, Pa. Therefore n = 0.5126 cycle per 
second; and substituting this value for n and the value 94.5 
centimeters for / in equation (18) we may calculate the value of 
g which comes out to be 980.18 centimeters per second per second. 
This result may be expected to be inaccurate for several reasons, 
namely, the observations themselves are somewhat inaccurate, 
the pendulum bob is not all at distance / from the point of sus- 
pension, the suspending thread or string has some mass, and the 
pendulum must swing through an appreciable amplitude when 
the observations for n are taken. 

This discussion of the simple pendulum is an approximation, 
because the amplitude of swing of the pendulum is assumed to he 
infinitely small which, of course it is not. In every other discus- 
sion in this chapter where we consider quantities " as small as 
you please " the results are rigorously exact. 



CHAPTER III. 

FRICTION. WORK AND ENERGY. 

39. Friction. — A moving body like a wagon or a boat is always 
acted upon by dragging forces which oppose its motion and tend 
to bring it to rest. This action of surrounding bodies on a body 
in motion is called friction. In many cases friction is extremely 
irregular, indeed there are only two cases in which it is sufficiently 
regular to admit of simple mathematical treatment as follows: 

Sliding friction. — When one smooth body slides over another 
we have what is called sliding friction. Thus the cross-head of 
an engine slides back and forth on the guides, and a rotating 
shaft slides in its bearings. The motion is in each case opposed 
by a frictional drag. 

Fluid friction. — The flow of water through a pipe is opposed 
by a frictional drag. This kind of friction is discussed in 
Chapter V. 

Coefficient of sliding friction. — A familiar experiment In 
elementary physics is to show that the horizontal force H 
required to drag a block along on a table is approximately pro- 
portional to the vertical force V which is pushing the block 
against the table, that is, the ratio H/V is a constant (approxi- 
mately) for a given pair of sliding surfaces, iron on brass, wood 
on iron, glass on wood, etc., and this ratio is called the coefficient 
of friction of the given pair of sliding surfaces. Thus the coeffi- 
cient of friction of smooth dry unvarnished wood on smooth 
metal is about 0.27. For example, a smooth 4-pound block of 
metal is placed upon a smooth dry horizontal board and loaded 
with an additional 6 pounds, making a total force of 10 "pounds" 
pushing the metal block against the board. It is then observed 
that a force of 2.y "pounds" is required to drag the block along 
the board. 

59 



60 MECHANICS. 

The force required to start sliding is always greater than the 
force required to maintain a steady sliding motion after it has 
been started. 

Very clean surfaces of like material tend to weld together 
thus making sliding friction extremely irregular. A good ex- 
ample is clean brass on brass, but the most striking example is 
clean glass on glass. 

The presence of water or oil in very small quantity sometimes 
complicates matters greatly. Thus two panes of glass cannot 
be made to slide over each other at all when there is a very 
small quantity of water between them. 

When smooth sliding surfaces are separated by a layer 
of oil or water of appreciable thickness, the force required to 
produce sliding is very small and it does not depend upon the 
force which pushes the sliding surfaces together except indirectly 
in that the force which pushes the sliding surfaces together tends 
to squeeze out the layer of oil or water between them. A fairly 
thick (viscous) oil is better than water or kerosene for lubrication 
for two reasons, namely, (a) the viscous oil is not squeezed out 
from between the sliding surfaces as easily as kerosene or water 
would be, and {h) a fresh supply of a viscous oil is continuously 
dragged into the region between the sliding surfaces. 

40. Active forces and inactive forces. Definition of work. — 

Nothing is more completely established by experience than the 
necessity of employing an active agent such as a horse or a steam 
engine to drive the machinery of a mill or factory, to draw a car, 
or to propel a boat; and although the immediate purpose of the 
driving force may be described in each case by saying that the 
driving force overcomes or balances the opposing forces of fric- 
tion, still the fact remains that the operation of driving a machine 
or propelling a boat involves a continued effort or cost. Indeed 
to supply a man with the thing (energy) which will drive his mill 
or factory, is to supply him with a commodity as real as the 
wheat he grinds or the iron which he fabricates into articles of 
commerce. Wheat and iron are sharply defined as commodities 



FRICTION. WORK AND ENERGY. 6 1 

in the popular mind on the basis of many generations of com- 
mercial activity, because wheat and iron can be stored up and 
taken from place to place, and because change of ownership is so 
easily accomplished and so simply accounted for. That which 
serves to drive a mill or factory, however, cannot be stored up 
except to a very limited extent, and it is only in recent years that 
means have been devised for transmitting it from place to place 
and that an exact system of accounting has been established for 
governing its exchange. A clear idea of energy does not exist 
as yet in the popular mind, and the following definitions cannot 
be expected to convey a full and clear idea at once. 

The common feature of every case in which motion is main- 
tained is that a force is exerted upon a moving body and in the 
direction in which the body moves. Such a force is called an 
active force,* and to keep up an active force requires continuous 
effort or cost. 

A force which acts on a stationary body, on the other hand, 
may he kept up indefinitely, without cost or effort; and such a 
force is called an inactive force. Thus a weight resting on a 
table continues to push downwards on the table, a weight sus- 
pended by a string continues to pull on the string, the main 
spring of a watch will continue indefinitely to exert a force upon 
the wheels of the watch if the watch is stopped. 

The idea of an inactive force is applicable also to a force which 
acts on a moving body but at right angles to the direction in 
which the body moves. Thus the vertical push of a driver on 
the seat of a wagon which travels along a level road is an inactive 

* An active force can be traced in every case to a substance which grows shorter 
under tension or to something which grows larger under compression. Thus a 
muscle under tension grows shorter, the steam in a steam engine cylinder grows 
larger or expands under pressure, and so on. 

According to the above definition the force with which a boy pulls on a sled is 
called an active force, but the boy does not move with respect to the sled, and so 
far as the relation between hoy and the sled is concerned, they might as well be aboard 
a train and both moving along together, or the boy might as well be pulling for- 
wards on the rear door knob of the car in which he is riding. The real activity is 
in the muscles of the boy's legs as, of course, everyone knows. 



62 MECHANICS. 

force, the forces with which the spokes of a rotating wheel pull 
inwards on the rim of the wheel are inactive forces. 

An active force is said to do work, and the amount of work 
done in any given time is equal to the product of the force and the 
distance that the body has moved in the direction of the force. 
That is 

W = Fd (19) 

in which F is the force acting on a body, and W is the work 
done by the force during the time that the body moves a dis- 
tance d in the direction of F. 

If ^ is not parallel to F, then W = Fd cos 6, where 6 is 
the angle between F and d. 

Units of work. — The unit of work is the work done hy unit force 
while the body on which the force acts moves unit distance in the 
direction of the force. 

The dyne-centimeter is the c.g.s. unit of work and it is the 
work done by a force of one dyne while the body upon which the 
force acts moves one centimeter in the direction of the force. 
The dyne-centimeter of work is called the erg. The erg is for 
most purposes an inconveniently small unit of work; therefore 
a multiple of the erg is extensively used, namely, the joule, which 
is equal to ten million ergs (lO'^ ergs). 

The foot-poundal is the f.p.s. unit of work and it is the work 
done by a force of one poundal while the body on which the force 
acts moves one foot in the direction of the force. The foot- 
poundal is seldom used in practice. 

The foot-'' pound.'' The ''pound" is the pull of gravity on a 
one-pound body in London, and the work done by this force while 
the body upon which it acts moves one foot in its direction is 
called a foot-'' pound.'' The foot-" pound" is extensively used 
as the unit of work among English-speaking engineers. 

41. Power. — The rate at which an agent does work is called the 
power of that agent. 

Examples. — A horse pulls on a plow with a force of 100 



FRICTION. WORK AND ENERGY. 63 

"pounds" and the plow moves 180 feet in the direction of the 
force in one minute. The work done is 180 feet X 100 "pounds" 
which is 18,000 foot-" pounds." Dividing the work done by the 
time required for doing it, namely, 60 seconds, we get 300 foot- 
" pounds" per second as the rate at which the horse does work. 

Units of power. — Power may of course be expressed in ergs per 
second or in foot-poundals per second, or in foot-" pounds" per 
second, as in the above example. The units of power which are 
most extensively used, however, are the watt and the horse-power. 

The watt is defined as one joule per second (ten million ergs 
per second). The kilowatt is 1,000 watts. 

The horse-power is defined as 746 watts, or 550 foot-" pounds" 
per second. 

42. Power developed by an active force. — Consider a force 
F acting on a body which moves in the direction of the force at 
velocity v. During t seconds the body moves through the 
distance vt, and the amount of work done is 7^ X vt, according 
to equation (19). Therefore, dividing the amount of work done, 
Fvt, by the time t during which it is done, we have the rate P 
at which the work is done, that is : 

P = Fv (20) 

If F is expressed in dynes and v in centimeters per second, 
then P is expressed in ergs per second. If 7^ is expressed in 
"pounds" and v in feet per second, then P is expressed in 
foot-" pounds" per second. 

Example. — A passenger locomotive exerts a pull of 6,000 
"pounds" on a train, and the velocity of the train is 90 feet per 
second. Therefore the net power developed by the locomotive 
(not counting the power required to propel the locomotive itself) 
is 540,000 foot-" pounds" per second, or 991 horse-power. 

Figure 49 shows a belt transmitting power from wheel A to 
wheel B. Let v be the velocity of the belt (which is also the 
velocity of the rims of both wheels). Let the pull of the belt 
on the rim of -B at & be -F and let the pull of the belt on the 



64 



MECHANICS. 



rim oi A at a be /. Then Fv is the rate at which work is 
done on the driven wheel B by the tight side of the belt, and 
similarly fv is the rate at which work is done on the driving 



driven 

wheel 

B 




driving 

wheel 

A 



Fig. 49. 

wheel A by the loose side of the belt (rate at which work is 
given back from driven wheel to the driving wheel). Therefore 
the net rate at which work is delivered to the driven wheel B is : 



or 



F = Fv - fv 
P = {F- f)v 



(21) 



Thus, for example, a belt travels at a velocity of 100 feet per 
second, the tension of the belt on the tight side is 1,000 ''pounds'* 
{= F), the tension of the belt on the loose side is 450 "pounds" 
{= f), and the net power transmitted by the belt is 550 "pounds " 
X 100 feet per second or 55,000 foot-" pounds" per second or 
100 horse- power. 

43. Power-time units of work. — Nearly all practical measure- 
ments relating to work are measurements of power, and it has 
therefore come about that a given amount of work done is often 
expressed as the product of power and time. 

The watt-hour is the amount of work done in one hour by an 
agent which does work at the rate of one watt. 

The kilowatt-hour is the amount of work done in one hour by 
an agent which does work at the rate of one kilowatt. 

The horse-power-hour is the amount of work done in one hour 
by an agent which does work at the rate of one horse-power. 



FRICTION. WORK AND ENERGY. 65 

ENERGY. 

44. Definition of energy. Limits of the present discussion. — 

Any agent which is able to do work is said to possess energy, and 
the amount of energy an agent possesses is equal to the total 
work the agent can do. Thus the spring of a clock when it is 
wound up is in a condition to do a definite amount of work, and 
it is therefore said to possess a definite amount of energy. 

In developing the idea of energy it is important to distinguish 
between an agent which merely transforms energy and an agent 
which actually has within itself the ability to do a certain amount 
of work. Thus the steam engine merely transforms the energy 
of the steam Into mechanical work, and a water wheel merely 
transforms the energy of an elevated store of water Into mechan- 
ical work, whereas a clock spring, when wound up, has a store of 
energy within itself. 

Whenever a substance (or a system of substances) gives up 
energy which it has In store, the substance always undergoes 
change. Thus the fuel which supplies the energy to a steam 
engine and the food w^hich supplies the energy to a horse, undergo 
chemical change; the steam, w^hich carries the energy of the fuel 
from the boiler to the engine, coo/5 0^ or undergoes a thermal change 
when it gives up its energy to the engine ; a clock spring changes its 
shape as it gives off energy ; an elevated store of water changes its 
position as it gives off energy; the heavy fly wheel of a steam 
engine does the work of the engine for a few moments after the 
steam is shut off and the fly wheel changes its velocity as it gives 
off Its energy. 

Not only does a substance undergo a change when It gives up 

energy by doing work, but a substance which receives energy or 

has work done upon it undergoes a change. Thus when air Is 

compressed by a bicycle pump, work is done on the air and it 

becomes warm; the work done in keeping up the motion of any 

machine or device produces heat at the places where friction 

occurs; when a clock spring Is wound up It stores energy by 

its change of shape ; when water is pumped into an elevated tank 
6 



66 MECHANICS. 

it stores energy by its change of position; a large part of the 
work which is expended on a heavy railway train at starting is 
stored in the train by its change of velocity. 

We now face a very important question; shall we attempt 
a complete discussion of the whole theory of energy at once by 
examining into all kinds of changes which take place when a 
substance does work or has work done upon it; or shall we 
base our discussion on one thing at a time? Most assuredly the 
latter. Therefore we proceed to discuss the energy relations 
involved in purely mechanical changes, namely, changes of 
position, changes of velocity, and changes of shape, and we 
exclude everjrthing else from our present discussion such as 
chemical changes and thermal changes. 

In attempting to exclude thermal changes from our present 
discussion, however, we are confronted by the fact that friction 
(with its accompanying thermal changes) is always in evidence 
everywhere; and it requires a very high degree of analytical 
power to think only of purely mechanical changes in the face of 
such a fact. This necessary feat of mental effort is greatly facili- 
tated by the use of the idea of a frictionless system; and this 
term will he used whenever it is desired to direct the reader's attention 
exclusively to the energy relations that are involved in purely 
mechanical changes. 

45. Kinetic energy and potential energy. — Before proceeding 
to a minute examination into the mechanical theory of energy, 
it is desirable to establish the ideas of kinetic energy and potential 
energy on the basis of general experience. Suppose that a post, 
standing beside a railway track, is to be pulled out of the ground ; 
can a car-load of stone be made to do the work? Certainly it 
can. All that is necessary is to have the car moving past the 
post and to throw over the post a loop of cable which is attached 
to the moving car. A moving car is able to do work; and when 
it does work its velocity is reduced, and its store of energy 
decreased. The energy which a body stores by virtue of its 
velocity is called the kinetic energy of the body. 



FRICTION. WORK AND ENERGY. 67 

It is also a familiar fact that a weight can drive a clock, but in 
doing so the position of the weight changes and its store of en- 
ergy is reduced. The energy which a body stores by virtue of its 
position is called the potential energy of the body. 

The physical reality which lies behind the terms kinetic energy 
and potential energy can perhaps be shown most clearly by con- 
sidering a bicycle rider. Suppose that the rider faces a steep hill 
or a sandy stretch of road where he is called upon to do an un- 
usual amount of work. Every bicycle rider realizes the ad- 
vantage of having a large velocity in such an emergency. This 
advantage of velocity is called kinetic energy. Or suppose that 
a bicycle rider wishes to use his whole strength, or more if he 
had it, in covering a certain distance. Every bicycle rider realizes 
the advantage of being on top of a hill in such an emergency. 
This advantage of position is called potential energy. 

46. Kinetic energy of a body in translatory motion. — The 

kinetic energy of a moving body is given by the equation : 

W = imv^ (22) 

in which W is the kinetic energy in ergs (or f oot-poundals) , 
m is the mass of the body in grams (or pounds) and v is the 
velocity of the body in centimeters per second (or feet per second). 

The kinetic energy of a moving body may be defined not only 
as the work it can do when stopped, but also as the work which 
must be done upon it to set it in motion. Therefore equation 
(22) can be established as follows: A constant unbalanced force 
F acts on a body of mass w, starting it from rest, and it is 
required to find the work which is done on the body while its 
velocity is increasing from zero to v. 

Let a be the acceleration of the body. Then: 

7^ = ma (i) 

according to Art. 6. The acceleration a is constant because 

F and m are both constant. Therefore the velocity v gained 

in t seconds is: 

V = at (ii) 



68 MECHANICS. 

and the distance d traveled by the body during the t seconds is : 

according to Art. i6. Therefore, multiplying equations (i) and 
(iii) member by member, we have : 

Fd = \maH'^ (iv) 

but Fd is the work W done on the body, and aH"^ is equal to 
v^, according to equation (ii), so that equation (iv) reduces to 
equation {22). 

47. Potential energy.^ — The energy which is stored in a body 
by virtue of its position has been called the potential energy of the 
body. It is impossible, however, to assign a definite amount of 
potential energy to a body in a given position. Thus the driving 
weight of a clock might have its available store of potential 
energy increased by boring a hole in the clock case so as to let 
the weight move down to the floor, then a hole could be bored in 
the floor, and eventually a well could be dug in the ground! 
In order to speak definitely of the potential energy of a body 
(with reference to the earth) it is necessary to choose an arbitrary 
zero position for the body,- and to reckon the potential energy 
of the body in any given position as the work the body can do in 
changing from the given position to the chosen zero position. 

Remark. — This definition of the potential energy of a body 
in a given position takes it for granted that the body can do a 
perfectly definite amount of work, no more and no less, as it 
moves from the given position to the chosen zero. position what- 
ever the path may be over which the body moves. 

48. The principle of the conservation of energy. Completion 
of the definition of potential energy. — The fact that energy can 
neither be created nor destroyed is called the principle of the 
conservation of energy. It is very desirable, however, to express 
this principle in experimental terms as in the following discussion, 
and when this is done we find that from the purely mechanical 



FRICTION. WORK AND ENERGY. 69 

point of view (thermal and chemical effects being absent) the 
principle of the conservation of energy grows out of the legitimacy 
of the definition of potendal energy. 

The moon is a body which stores potential energy by virtue 
of its position relative to the earth. Let us therefore consider 
the earth and moon, let us assume that no forces from outside act 
upon either, and let us take the earth as our reference base so 
that we need think of the moon, only, as moving. The arrow F 
in Fig. 50 represents the force with which the 
earth acts on the moon, and when the moon moon^J^A 
moves from ^ to ^ the force F does a 1 

certain amount of work W. Concerning this X K B ) 

amount of work W two things may be said, 
namely, (a) it is, by definition, equal to the 
decrease of potential energy of the moon as the 
moon moves from A to B] and {h) it can be 
shown by an argument similar to Art. 46 that eairth\ 
the increase of kinetic energy of the moon while 
it moves from ^ to 5 is equal to W. There- pj ^^ 

fore the total energy of the moon and earth 
can never change, outside forces being assumed not to exist, 
such forces, for example, as might be exerted on earth and moon 
by the sun and other planets. 

What is here said concerning the earth and moon is true of 
any system of things which exert forces on each other but which 
are free from actions which produce heat. Such an ideal system 
is called a purely mechanical system, and the total energy 
{potential energy plus kinetic energy) of such a system (force 
action from outside the system being assumed not to exist) 
n&uer changes, that is, if our definition of potential energy is 
legitimate. But our definition of potential energy is not legiti- 
mate until we know by experiment that the amount of work 
done by the internal forces of a purely mechanical system while 
the system is changing from one condition to another condition 
is the same whatever the intermediate stages may be through 




70 MECHANICS. 

which the system passes in being brought from one condition 
to the other. If this were not true it would be meaningless to 
speak of the potential energy of the system in a given condition 
or state. Indeed the above statement is known by experiment 
to be true, and the statement is the essence of the principle 
of the conservation of energy in its purely mechanical sense. 
When reduced to its simplest terms, therefore, the principle of the 
conservation of energy is that you cannot get more work out of a 
system hy letting it down, as it were, from condition A to condition 
B than must he done to carry the system hack from B to A. 

Take a stone and lower it from a high position ^ to a low 
position B, thus getting work or energy out of it; then bring 
the stone back from B to A, slipping it behind your back in the 
hope or pretense of getting it back to A with a small expendi- 
ture of work! This procedure strongly suggests the point of 
view of every perpetual-motion promoter, namely, a vague and 
wholly unintelligent expectation of success or downright cheating! 

Remark. — The extension of the principle of the conservation 
of energy to include heat effects is discussed in Chapter VII of 
Part II. 



CHAPTER IV 



HYDROSTATICS. 

49. Hydrostatic pressure. — A fluid* at rest always pushes 
normally against a surface which is exposed to the action of the 
fluid. Thus the small arrows in Fig. 51 show how the steam in 
a steam-engine cylinder pushes against the piston, the arrows in 
Fig. 52 show how the water in a pail pushes against the sides of 



open 



-^ 



piston 



steam 




cylinder 

Fig.' 51. 

the pail (of course the water pushes on the bottom of the pail 
also), and the arrows in Fig. 53 show how the water in a tank 
pushes against the surface of a submerged ball. 

A fluid at rest not only pushes on a surface which is exposed 
to its action, but two contiguous portions of the fluid push on 
each other as shown in Fig. 54. The dotted line qq represents 
a small plane area or surface which may be imagined to separate 
two contiguous portions of the fluid, and the small arrows show 
how the portions of the fluid push on each other. 

A fluid in motion does not necessarily push normally against 
a surface which is exposed to the action of the fluid. Thus the 
small arrows in Fig. 55 show how the moving water in a pipe 
pushes against the walls of the pipe. Also, when a fluid is 
moving the forces which are indicated by the arrows in Fig. 54 
are not necessarily at right angles to the plane area qq. Thus 

* The term fluid includes both liquids and gases. 

71 



72 



MECHANICS. 



the small arrows in Fig. 56 show how the portions of the moving 
water on the two sides of the plane gg push on each other 



open 





across qq. The remainder of this chapter deals with fluids at rest. 

The force F with which a fluid at rest pushes against an element^ 
of an exposed surface is at right angles to the element as above 
stated, and it is proportional to the area a of the element; and 
the force per unit area, F/a, is called the hydrostatic pressure or 
simply the pressure of the fluid at the place where the element of 
area is located. Representing the pressure of a fluid at a point 
by p, we have 




pipe 

TTTTTTT7 



pipe 



direction of motion 
of water 



Fig. 55- 




Fig. 56. 



F 

p=- 

a 



or 



F = pa (23) 

A simple force exerted on an object is sometimes called a 
pressure." In this text, however, the word pressure always 

* Meaning a small portion of the exposed surface. 



HYDROSTATICS. 73 

means force per unit area, and the force is understood to be at 
right angles to the area. 

50. Uniform pressure and non-uniform pressure. — The pres- 
sure of the air in a room is everywhere very nearly the same in 
value, about 15 "pounds" per square inch (about 480 poundals 
per square inch or 1,000,000 dynes per square centimeter). 
The pressure of a fluid is said to be uniform in a region where 
the pressure has everywhere the same value. 

The water in a pail pushes with greater and greater force 
upon each unit area of exposed surface at greater and greater 
depths as indicated by the arrows. Figs. 52 and 53. That is to 
say, the pressure of the water increases with depth. The pressure 
of a fluid is said to be non-uniform in a region where it does not 
have everywhere the same value. 

When the pressure of a fluid is uniform, the force exerted on 
any plane area can be calculated by equation (23). Thus the 
piston in Fig. 51 has an area of 300 square inches, the steam 
pressure is 100 "pounds" per square inch and the total force 
exerted on the piston is 100 "pounds" per square inch X 300 
square inches or 30,000 "pounds." 

When the pressure of a fluid is non-uniform equation (23) 
must be written : 

AF = p'Aa (24) 

where AF is the force exerted by the fluid on a very small 
plane area Aa. 

51. Pascal's principle. — The force I^F with which a fluid 
pushes against a small area Aa varies, of course, with the 
location of Aa if the pressure of the fluid is non-uniform, but 
the force AF is the same whatever the direction of Aa {at a given 
place) j^ This proposition is known as Pascal's principle from 
its discoverer, and it can be proved theoretically from the proposi- 
tion of normal force action as given in Art. 49, a simple experi- 
mental demonstration must, however, suffice. Figure 57 shows 

* Fluid being at rest. 



74 



MECHANICS. 



a glass cup C over the mouth of which a thin sheet of rubber is 

tied. The air is partly drawn ouc 
of the cup so that the rubber sheet 
is pushed inwards by the outside 
air. Then, with the stop-cock 
closed, the dished shape of the rub- 
ber sheet remains unchanged as the 
cup is turned so as to make the 
rubber sheet face upwards or down- 
wards or sidewise. 

52. Circumferential tension in the walls of a cylindrical pipe. 

— The pressure of a fluid in a pipe produces a state of tension in 
the walls of the pipe. Figure 58 is a side view of a pipe of radius 
r. Consider a narrow band of the pipe abd of width w. An 
end view of this band is shown in Fig. 59. Imagine the fluid in 




Fig. 57- 




pipe 



axis of 
pipe 





d 
side view 

Fig. 58. 

the bottom half of the pipe to be solid as indicated by the shading 
in Fig. 59, and let us consider the shaded half of the band 
of pipe (together with its contents) as a definite body B. 
This body B is stationary. Therefore the forces which act 
upon B are balanced. But the fluid in the upper half of the 
pipe pushes downwards on B with a force equal to p X w X 2r 
as indicated by the vshort arrows, w X 2r being the area of the 
upper face of B, and p being the pressure of the fluid. The 
band of pipe must, therefore, pull upwards on B to balance this 
downward push of the fluid. Indeed the two arrows FF repre- 



HYDROSTATICS. 75 

sent the two equal forces with which the band pulls up on B. 

Therefore : 

2F = 2rwp (i) 

What is called the circumferential tension in the pipe is the force 

per unit width of the band, namely, F/w. Therefore, from 

equation (i) we have: 

F 
circumferential tension = — — rp (25) 

w 

Example. — A steam pipe Is 8 inches in diameter (4 Inches in 
radius) and the pressure of the steam in the pipe is 100 "pounds" 
per square inch (3216 poundals per square inch). Therefore the 
circumferential tension of the material of the pipe is 4 inches 
multiplied by 100 "pounds" per square inch which gives 400 
"pounds" per inch. That is, each band of the pipe one inch in 
width is acted upon by a stretching force of 400 "pounds." 

53. Pressure in a liquid due to gravity. — The pressure in a 
fluid under the action of gravity increases with the depth. If the 
density of the fluid is the same throughout, (and this is approxi- 
mately the case in any liquid), then the pressure at a point distant 
X beneath the surface of the liquid exceeds the pressure at the 
surface by the amount 

p = xdg (26) 

in which p is expressed in dynes per square centimeter when 
the density d of the liquid is expressed in grams per cubic centi- 
meter, the distance x in centimeters and the acceleration of 
gravity g in centimeters per second per second ; or p is expressed 
in poundals per square foot if the density of the liquid d is ex- 
pressed in pounds per cubic foot, the distance x in feet and 
the acceleration of gravity g in feet per second per second. 

The most useful form of the above equation is that which gives 
the pressure in "pounds" per square foot, namely 

p = xd (27)* 

* This equation is strictly true only in London because elsewhere the pull of 
the earth on a one-pound body is not exactly what we have defined as a "pound" 
of force. See Arts. 5 and 6. 



76 MECHANICS. 

in which p is expressed in " pounds " per square foot, x is ex- 
pressed in feet and d is the density of the Hquid in pounds per 
cubic foot. 

Discussion of equation (26). The force with which a Uquid 
pushes on an element of an exposed surface is independent of the 
direction of the element according to Pascal's principle. There- 
fore we may derive equation (26) by considering a horizontal 
surface a square feet in area exposed to the action of the liquid 
as shown in Fig. 60. The volume of the liquid directly above 

snrikce ofllqtdd- 





»_-- _» ^ 



Fig. 60. Fig. 61. 

a is ax cubic feet, the mass of this portion of liquid is axd pounds 
(where d is the density of the liquid in pounds per cubic foot), 
the force in poundals with which gravity pulls on this portion of 
the liquid is axdg, and therefore the total force with which this 
portion of liquid pushes down on the element a is equal to axdg 
poundals, so that the force per unit area is axdg divided by a, 
or xdg poundals per square foot. 

Equation (26) involves no consideration of the shape of the 
vessel which contains the liquid. As a matter of fact, the pres- 
sure at a point in a liquid exceeds the pressure at the surface of 
the liquid by the amount xdg whatever the shape and size of 
the containing vessel may be. This may be made almost self- 
evident as follows: Given a point p, Fig. 61, at a distance x 
beneath the surface of a large body of liquid. Imagine a por- 
tion of the liquid A AAA, of any shape whatever, extending from 
p to the surface. The liquid surrounding the portion AAAA 



HYDROSTATICS. 77 

acts on AAAA exactly as a containing v^essel of the same shape 
would act, and therefore the pressure of p is exactly what it 
would be if the portion AAAA were contained in such a vessel. 

54. Atmospheric pressure. The barometer. — The force with 
which the air pushes on the surfaces of bodies does not ordi- 
narily appeal to our senses. It is shown, however, by the 
collapse of a thin-walled vessel when the air is pumped out of the 
vessel. Atmospheric pressure is also shown by means of what 
are called the Magdeburg hemispheres. Two metal cups fit to- 
gether air tight, and when the air is pumped out of the enclosed 
region a very considerable force is required to pull the cups apart. 
This celebrated experiment was devised by Otto von Guerike, 
the inventor of the air pump, and it was performed publicly at 
Magdeburg in 1654. 

The barometer is an instrument for measuring the pressure of 
the air. It consists of a tube 7", Fig. 62, filled with mercury and 
inverted in an open vessel of mercury as shown. The tube T 
is about a yard in length, and an empty space V is left in which 
the pressure is zero.* 

The pressure of the mercury in the tube at the level of the outside 
mercury surface is equal to the pressure of the air, and it exceeds 
the pressure in V by the amount xdg, according to equation 
(26) where x is the height of the mercury column in the tube as 
shown in the figure, d is the density of the mercury, and g is 
the acceleration of gravity. This expression xdg gives the 
value of the atmospheric pressure in dynes per square centimeter 
(or in poundals per square foot) as explained in Art. 53. 

If the mercury is at some standard temperature, d is in- 
variable; and if the barometer is used in a given locality, g is 
invariable ; and under these conditions the distance x may he used' 
as a measure of the pressure. In fact, atmospheric pressure is 
usually expressed In terms of the height the barometric column 
would have in millimeters or in inches if the mercury were at 

* Even if the tube is filled with great care (mercury boiled in the tube as it is 
filled), mercury vapor will form in the region V and the pressure in this region 
will not be quite zero. 



78 



MECHANICS. 




c;;--i 



air. 



\0 



0° C. and if the value of the acceleration of gravity were 981.61 

cm./sec.^ (its value at 45° north latitude at sea level). To 

facilitate the accurate use of the bar- 
ometer in different localities and at 
different temperatures, tables* have 
been published, with the help of which 
the height of barometric column under 
standard conditions as to temperature 
and gravity may be easily found from 
its observed height under known con- 
ditions. 

The aneroid barometer consists of an 
air tight capsule or box with a thin 
metal lid which bends more or less as 
the pressure of the outside air varies; 
this variable bending actuates a 
pointer which plays over a divided 
scale, and the scale is calibrated to 
give true values of the atmospheric 

pressure. The aneroid barometer is extensively used as a field 

instrument because of its easy portability. 

55. Manometers or pressure gauges. — The barometer is used 
for measuring the pressure of the atmosphere. An instrument 
for measuring the difference between atmospheric pressure and 
the pressure in a closed vessel like a steam boiler or water tank 
is called a manometer or a pressure gauge. The most extensively 
used manometers are the so-called open tube manometer, and the 
Bourdon gauge. 

The open tube manometer. — Figure 63 shows an open tube 
manometer mm arranged for measuring the pressure of the gas 
in a gas main (amount by which the gas pressure exceeds atmos- 
pheric pressure). The difference in level h of the liquid in the 
two arms of the glass tube mm is measured, and the pressure of 

* To be found in many laboratory reference books, and in the hand book of the 
United States Weather Bureau. 



^//////m'//jy//>y///x//xyy'///y/MMli. 



Fig. 62. 



HYDROSTATICS. 



79 



the gas (amount by which gas pressure exceeds atmospheric 
pressure) is equal to hdg, where d is the density of the Uquid in 
the tube, and g is the acceleration of gravity (see Art. 53). 

The Bourdon gauge consists of a very thin-walled metal tube, 
TT, Fig. 64, which is supported at end A , and which is connected 

open 



gas main 



m 



T 
I 
I 

\h 

I 
I 
I 

.A 



m 




Fig. 63. 



Fig. 64. 



at p to the region in which the pressure is to be measured. 
Variations of pressure (in the tube TT) cause the free end B of 
the tube to move, and this movement actuates a pointer which 
plays over a divided scale. 

56. Buoyant force of fluids. — The pressure in a fluid increases 
with depth as explained in Art. 53. Therefore the upward 
forces exerted on the lower surface of a submerged body are 
greater than the downward forces exerted on the upper surface 
of the body as shown in Fig. 53. Therefore, on the whole, a 
fluid exerts an upward force on a submerged 
body. This upward force is called the buoyant 
force of the fluid on the body. 

Archimedes' principle. — The buoyant force 
exerted by a fluid on a submerged body of 
volume V is equal to the weight of the same 
volume of the fluid. This fact is called Archi- 
medes^ principle from its discoverer, and it 
may be made almost self-evident from the fol- 
lowing considerations. Given a fluid at rest, 
and let us think of a definite portion A A of this fluid of any size and 



\B 



fluid -. — 




— -- fluid 



W 



Fig. 65. 



8o MECHANICS. 

shape, as shown in Fig. 65. This portion A A is stationary, and 
therefore the downward pull of gravity on A A (the weight W 
of A A) must be balanced by the total* force B with which the 
surrounding fluid pushes upwards on AA. That is B must be 
equal and opposite to W, and B and W must lie in the same 
straight line. But the surrounding fluid pushes on AA exactly 
as it would on any submerged body of the size and shape of AA. 
Therefore B is the buoyant force of the fluid on any body of the 
size and shape of AA, and as stated above, B is equal in value 
to the weight W of the portion AA of fluid. 

Archimedes' principle as applied to a body floating on a liquid. 
— The buoyant force of a liquid on a floating body is evidently 
equal (and opposite) to the downward pull of gravity on the body 
(the weight of the body) because the floating body remains 
stationary. But the buoyant force of a liquid on a floating 
body is equal to the weight of a volume v of the liquid, where v 
is the volume of the portion of the body which is below the surface 
of the liquid. Therefore a floating body displaces its weight of the 
liquid in which it floats. 

Examples. — A balloon has a total volume of 400 cubic meters, 
and the balloon and contained hydrogen have a mass of 286 
kilograms. The balloon, of course, displaces 400 cubic meters of 
air, and the density of air is i .2 kilograms per cubic meter, so that 
the balloon displaces 480 kilograms of air. Therefore the total 
buoyant force of the air on the balloon is equal to the weight of 
480 kilograms of material, and the downward pull of the earth 
on the balloon is of course equal to its weight (the weight of 
286 kilograms of material). Consequently the upward push of 
the air on the balloon exceeds the weight of the balloon by an 
amount equal to the weight of 480 kilograms minus 286 kilograms 
or 194 kilograms of material. 

A boat displaces 2,000 cubic feet of water so that the buoyant 
force of the water on the boat (which is equal to the weight of 

* The fluid exerts a push on each element of the surface oi AA, and these pushes 
are together equivalent to the single force B which is called their vector sum 
or resultant. 



HYDROSTATICS. 8l 

the boat) is equal to the weight of 2,000 cubic feet of water, or 
to the weight of 125,000 pounds of water. 

57. Density and specific gravity. — The density of a substance 
is its mass per unit volume, that is 

^ = f (28) 

where M is the mass of a body in grams (or pounds), V is 
its volume in cubic centimeters (or cubic feet) and D is its den- 
sity in grams per cubic centimeter (or in pounds per cubic foot). 

The specific gravity of a substance at a given temperature is the 
ratio of the density of the substance to the density of water at 
the same temperature. Thus, to say that the specific gravity 
of iron is 7.78 means that the density of iron is 7.78 times the 
density of water, or it is equivalent to saying that the mass of a 
given volume of iron is 7.78 times the mass of the same volume 
of water. 

In some cases a great deal of tedious circumlocution is required 
to distinguish sharply between weight and mass. The pull of 
the earth on a one-gram body (or on a one-pound body) at any 
given place is a perfectly definite force, and if we express the 
weight of a body in terms of this unit of force, then the weight of 
a body will be the same thing numerically as its mass. This 
simple scheme is adopted in the following discussion. 

(a) The volume of a vessel can be very accurately determined 
by weighing the empty vessel and then weighing the vessel full 
of water or mercury. The difference is the net weight of the 
water or mercury, and by dividing this by the density of the 
water or mercury the volume of the vessel is found. Example. 
A vessel weighs 286.52 grams empty and 948.23 grams when 
filled with water at 20° C. Therefore the net weight of the 
water is 661.71 grams, which divided by 0.9983 grams per cubic 
centimeter (the density of water at 20° C.) gives 663 cubic 
centimeters as the volume of the vessel. 

(6) A substance having a volume of V cubic centimeters 
7 



82 MECHANICS. 

weighs 546.2 grams in air* and 432.6 grams when suspended by 
a thread and submerged in water at 20° C. The difference is 
the buoyant force of the water, which is the weight of V cubic 
centimeters of water. Therefore the weights of equal volumes 
of the substance and of water are 546.2 grams and 113.6 grams 
respectively, so that the specific gravity of the substance at 20° 
is 546.2 grams -^ 113.6 grams or 4.808. That is, the substance 
is 4.808 times as heavy as water. But the density of water at 
20° C. is 0.9983 gram per cubic centimeter, and therefore the 
density of the substance is 4.808 X 0.9983 grams per cubic 
centimeter or 4.6998 grams per cubic centimeter. 

(c) A glass ball weighs 72.44 grams in air, 45.22 grams in 
water, and 47.94 grams in oil. The loss of weight of the ball in 
water (27.22 grams) is the weight of its volume of water, and the 
loss of weight of the ball in the oil (24.50 grams) is the weight of 
its volume of oil. Therefore the specific gravity of the oil is 
24.50 grams -^ 27.22 grams or 0.900. That is, the oil is 0.900 
as heavy as water. 

58. Cohesion ; adhesion. — When a body is under stress, as for 
example a stretched wire, the tendency of the stress is to tear 
the contiguous parts of the body asunder. The forces which 
oppose this tendency and hold the contiguous parts of a body 
together are called the forces of cohesion. The forces which 
cause dissimilar substances to cling together are called the forces 
of adhesion. The cohesion of water and the adhesion between 
water and glass are the forces which determine the curious 
behavior of water in a fine hair-like tube of glass, and the phe- 
nomena exhibited by liquids because of cohesion and adhesion 
are called capillary phenomena from the Latin word capillaris 
meaning hair. 

59. Surface tension. — On account of their cohesion, all liquids 
behave as if their free surfaces were stretched skins, that is, as if 
their free surfaces were under tension. Thus a drop of a liquid 
tends to assume a spherical shape on account of its surface 

* Buoyant force of air is neglected in these examples. 



HYDROSTATICS. 83 

tension. A mixture of water and alcohol may be made of the 
same density as olive oil, and a drop of olive oil suspended in 
such a mixture becomes perfectly spherical. 

Many curious phenomena* are produced by the variation of 
the surface tension of a liquid with admixture of other liquids or 
with temperature. Thus a drop of kerosene spreads out in an 
ever widening layer on a clean water surface, on account of the 
fact that the tension of the clean water surface beyond the layer 
of oil is greater than the tension of the oily surface. A small 
shaving of camphor gum darts about in a very striking manner 
upon a clean water surface, on account of the fact that the 
camphor dissolves in the water more rapidly where the shaving 
happens to have a sharp projecting point, the water surface has 
a lessened tension where the camphor dissolves, and the greater 
tension on the opposite side pulls the shaving along. A minute 
cork boat with a small bit of camphor gum fixed to its stern is 
pulled along for the same reason. A thin layer of water on a 
horizontal glass plate draws itself away and leaves a dry spot 
where a drop of alcohol is let fall on the plate. A thin layer 
of lard on the bottom of a frying pan pulls itself away from the 
hotter parts of the pan and heaps itself up on the cooler parts, 
because of the greater surface tension of the cooler lard. 

60. Angles of contact. Capillary elevation and depression. — 

The clean surface of a liquid always meets the clean walls of a 
containing vessel at a definite angle. Thus a clean surface of 
water turns upwards and meets a clean glass wall tangentially, 
and a clean surface of mercury turns downwards and meets a 
clean glass wall at an angle of 51° 8'. 

Since a clean water surface turns upwards and meets a glass 
wall tangentially it is evident that the surface of water in a small 
glass tube must be concave as shown in Fig. 66, and the result is 
that the water is drawn up into the tube by the surface tension. 

* See the very interesting article capillary action in the 9th edition of Encyclo- 
pedia Britannica. This article also gives a comprehensive discussion of the theory 
of capillary action. 



84 MECHANICS. 

On the other hand, the surface of mercury in a small glass tube 




Fig. 66. 




Fig. 67. 



Is convex and the surface tension pulls the mercury down below 
the level of the surrounding mercury as shown in Fig. 67. 



CHAPTER V. 

HYDRAULICS. 

6i. Subject and limitations of this chapter. — Hydraulics, in 
the general sense in which the term is here used, is the study of 
liquids and gases in motion; and the phenomena which are pre- 
sented in this branch of physics are excessively complicated. 
Even the apparently steady flow of a river through a smooth 
sandy channel is an endlessly intricate combination of boiling 
and whirling motion; and the jet of spray from a hydrant, or the 
burst of steam from the safety-valve of a locomotive, what is to 
be said of things such as these? Or let one consider the fitful 
motion of the wind as indicated by the swaying of trees and 
as actually visible in driven clouds of dust and smoke, or the 
sweep of the flames in a conflagration ! These are actual examples 
of fluids in motion, and they are indescribably, infinitely* com- 
plicated. 

The science of hydraulics is based on ideas which refer to average 
aspects of fluid motion. Thus the engineer is concerned chiefly 
with such things as the time required to draw a pail of water 
from a hydrant, the loss of pressure in a line of pipe between a 
pump and a fire nozzle, or the force exerted by a water jet on the 
buckets of a water wheel. These are called average effects be- 
cause they are never perfectly steady but always subject to per- 
ceptible fluctuations of an erratic character, and to think of any 

* Everyone concedes the idea of infinity which is based upon abstract numerals 
(one, two, three, four and so on ad infinitum I), and the idea of infinity which is 
based on the notion of a straight Hne; but most men are wholly concerned with the 
humanly significant and persistent phases of the material world, their perception 
does not penetrate into the substratum of utterly confused and erratic action which 
underlies every physical phenomenon, and they balk at the suggestion that the 
phenomena of fluid motion, for example, are infinitely complicated. Surely the 
abstract idea of infinity is as nothing compared with the intimation of infinity that 
comes from things that are seen and felt, 

85 



86 



MECHANICS. 



of these effects as having a definite value Is, of course, to think of 
its average value under the giveii conditions. The extent to 
which the practical science of hydrauKcs is Umited by the con- 
sideration of average effects is evident from the following outline 
of the ideal types of flow upon which nearly the whole of the 
science is based. 

Permanent and varying states of flow. — When a hydrant is 
suddenly opened, it takes an appreciable time for the flow of 
water to become steady. During this time (a) the velocity at each 
point of the stream is increasing and perhaps changing in direc- 
tion also. After a short time, however, the flow becomes fully 
established and then ih) the velocity at each point in the stream re- 
mains unchanged in magnitude and direction.* The motion {a) 
is called a varying state of flow, and the motion {h) is called a 
permanent state of flow. Most of the following discussion applies 
to permanent states of flow, indeed there are but few cases in 
which it is important to consider varying states of flow. 

The idea of simple flow. Stream lines. — ^The idea of simple 
flow applies both to permanent and to varying states of flow, but 
it is sufficient to explain the idea in its application to permanent 
flow only. When water flows steadily through a pipe, the motion 
is always more or less complicated by continually changing 




eddies, the water at a given point does not continue to move in a 
fixed direction at a constant velocity; nevertheless, it is conve- 
nient to treat the motion as if the velocity of the water were in 
a fixed direction and of constant magnitude at each point. Such 
a motion is called a simple flow. In the case of a simple flow, a 



* Assuming the stream to be free from turbulence, 
of simple flow. 



See the following definition 



HYDRAULICS. 



87 



line can he imagined to he drawn through the fluid so as to he at 
each point in the direction of the flow at that point. Such a line 
is called a stream line. Thus the fine lines in Fig. 68 are stream 
lines representing a simple flow of water through a contracted 
part of a pipe. To apply the idea of simple flow to an actual 
case of fluid motion is the same thing as to consider the average 
character of the motion during a fairly long interval of time. 

Lamellar flow. — Even though the motion of water in a pipe 
may be approximately a simple flow, the velocity may not be the 
same at every point in a given cross-section of the pipe, that is, 
the velocity may not be the same at every part of the layer ah, 
Fig. 69; in fact the water near the walls always moves slower 




than the water near the center of the pipe; nevertheless, it is con- 
venient in many cases to treat the motion as if the velocity were 
the same at every point in any layer like ah, Fig. 69. Such an 
ideal flow is called a lamellar flow, because in such a flow the 
fluid in any layer or lamella ah would later be found in the layer 
cd, and still later in the layer ef. To apply the idea of lamellar 
flow to an actual case of fluid motion is the same thing as to con- 
sider the average velocity over the entire cross-section of a 
stream. 

62. Some phenomena of fluid motion not associated with 
permanent, simple, lamellar flow. — The theoretical treatment 
of fluid motion in this chapter is so largely based on the ideas of 
permanent, simple, lamellar flow that we shall be carried far 
away from many interesting phenomena of fluid motion. 

The action of the hydraulic ram is a good example of varying 
flow. Water flows from a low dam through a long pipe PP and 



88 



MECHANICS. 



escapes through an open valve A as indicated in Fig. 70. The 
flowing water lifts the valve A thus suddenly closing it, and 



to high level tank 



water 




Fig. 70. 

the hammer or ram action of the moving water in PP driv^es a 
small quantity of water through the check valve B and into 
a high level tank. The water in PP is thus brought to rest with 
excessive pressure in CA which causes the column of water 
PPCA to rebound thus very greatly reducing the pressure in CA 
and opening the valve A as at first, and the above described 
action is repeated. 

The tendency of a stream of fluid to become turbulent (to 
depart from ideal simple flow) is exemplified by the sensitive 
flame. — When a fluid flows slowly through a channel or pipe, 
or as a jet through a body of surrounding stationary fluid, the 
motion approximates very closely to a simple flow. When the 
velocity of the fluid is increased, however, a critical velocity is 
soon reached at which the flow suddenly becomes very turbulent 
(full of eddies). This sudden increase of turbulence is illustrated 
by the behavior of an ordinary gas flame (the flame serves only 
to make the jet of gas visible). When the gas is turned on slowly 
the flame is at first smooth and steady, but a certain point is 
reached (a certain velocity of the gas in the jet) at which the 
flame suddenly becomes lough and unsteady, innumerable eddies 
form at the boundary between the moving gas and the still air. 
When the flame is on the verge of becoming unsteady it is some- 
times extremely sensitive, the least hissing sound causes it to 
become turbulent. 

The boundary between the moving gas and the still air con- 
stitutes what is called a vortex sheet, and the behavior of the 



HYDRAULICS. 



89 



sensitive flame is due to the instability of a vortex sheet; any 
disturbance, however small, starts a minute eddy which develops 
more and more. 

The behavior of the so-called spit ball is due to the instability 
of the vortex sheet. — A perfectly smooth spherical ball (not 
spinning) moves forwards through still air, and, since everything 
is symmetrical with respect to the line of motion, there can be no 
reason why the ball should jump to the right rather than to the 
left, therefore we may conclude that the ball will not jump 
either way ! But the ball does j ump side- 
wise as in case of the so-called spit hall, 
and a marble dropped in a jar of water 
follows an irregular zigzag path as indi- 
cated by the dotted Hne in Fig. 71. 

A sharp pointed stick stands vertically 
in a still room, and, since everything Is 
symmetrical with respect to the axis of 
the stick, there can be no reason why 
the stick should fall one way rather than 
another, therefore we may conclude that 
the stick will not fall either way! But 
the stick does fall. The vertical stick is 
In a condition of Instability, and any 

disturbance, however small, starts the stick falling, and a fall once 
started develops more and more, as every one knows. 

Figure 72 shows the air blowing past a ball (as If the ball were 
stationary and the air moving) and breaking away from the ball 
so as to leave a body of still air back of the ball. We thus have a 
vortex sheet ao^ and hh, and If the velocity of the air (velocity of 
the ball If the air is still) Is just right this vortex sheet Is unstable. 
Any disturbance, however small, starts an eddy which develops 
more and more. As a result there Is a sidewlse fluttering of the 
air stream back of the ball, and the reaction of this fluttering 
stream pushes the ball to one side and to the other Irregularly. 
This effect is familiar to anyone who has held a thin stick In the 




90 



MECHANICS. 



moving water at one side of a row boat or launch, the fluttering 
vortex sheets back of the stick react on the stick and give to it 
a quivering motion. 

The distinction between lamellar flow and non-lamellar flow 
in a pipe or channel is a simple example of a more fundamental 
distinction between irrotational flow and rotational flow, between 
a kind of flow in which the individual particles of the fluid do 




Fig. 72. 

not rotate and a kind of flow in which the individual particles 
of the fluid do rotate. This distinction is one of great im- 
portance in the mathematical theory of fluid motion.* The 
familiar smoke ring and the whirlpool in an emptying wash 
bowl are examples of rotational fluid motion. 

When water flows out of a hole in the bottom of a bowl a 
whirlpool generally forms above the hole. The formation of this 
whirlpool depends upon the previous existence of a slow rotatory 
motion of the water in the howl, which rotatory motion is greatly 
increased when the water flows towards the hole. This increase 
of spin velocity of a body as the parts of the body move towards 
the axis of spin is strikingly illustrated by the following experi- 
ment. A person standing on a pivoted stool is set spinning 
about a vertical axis with his arms outstretched and with weights 
in his hands ; and when he draws the weights in towards his body 
(towards the axis of spin) his spin velocity is greatly increased. f 

* See Franklin, MacNutt and Charles' Calculus, pages 242-250. 

t The spin-momentum Ks of a rotating body never changes unless an outside 
torque acts on the body. Therefore, if the spin-inertia K of the rotating body is 
decreased, its spin velocity 5 must increase. 



HYDRAULICS. 9^ 

The rotation of the earth on its axis involves a slow turning 
of one's horizon about a vertical axis (except at the equator). 
When the air near the ground is warmed by the sun's rays it 
starts to flow upwards at some place, and a chimney-like effect 
is produced by the rising column of warm air, the lower layer 
of warm air is drawn towards this central chimney from all 
sides and the slow turning motion of one's horizon becomes 
greatly increased as a more or less violent whirl at the cen- 
tral chimney. The cyclone"^ is a movement of this kind cover- 
ing thousands of square miles of country with a central chimney 
hundreds of miles in diameter. The tornado is a movement 
of this kind covering only a few square miles of country with a 
central chimney only a few hundred yards in diameter. The 
tornado is often very violent and destructive. 

63. Rate of discharge of a stream. — The volume of water 
which is delivered per second by a stream is called the discharge 
rate of the stream. Thus the mean discharge rate of the Niagara 
River is 300,000 cubic feet per second. The rate of discharge 0} a 
stream is equal to the product of the average velocity, v, of the stream 
and the sectional area, a, of the stream. For example, let PP, 
Fig. 73, be the end of a pipe out of which water is flowing, and 
let us assume that the velocity of flow has the same value v over 
the entire section of the stream (lamellar flow), then the water 
which flows out of the end of the pipe in t seconds would make 
a cylinder or prism of length vt, and of sectional area a, as 
indicated in the figure, and the volume of this water is therefore 
avt. Dividing this volume by the time t gives the discharge 
rate av. 

Variation of velocity with sectional area of a steady lamellar 
stream. Consider a simple flow of water through a pipe as indi- 
cated by the stream lines in Fig. 68. Let a' and a'^ be the 
cross-sectional areas of the stream at any two points P' and 
P"y and let v' and v'^ be the average or lamellar velocities of 
the stream at P' and P" respectively. Then a'v' is the 

* What is popularly called a cyclone is properly called a tornado. 



92 MECHANICS. 

volume of water which passes the point P' per second, and 
a"v" is the volume of water which passes the point P" per 




Fig. 73- 

second; and, therefore, since the same amount of water must 
pass each point per second, we have 

aV = a'V (29) 

that is, the product av has the same value all along the pipe, 
so that V is large where a is small, and v is small where a is 
large. 

Equation (29) applies only to a fluid which is approximately incompressible 
like water or any other liquid. In such a case a'v' is the amount of water per 
second entering one end of a pipe and a"v" is the amount of water per second 
flowing out of the other end of the pipe, and these two expressions must be equal 
to each other. If, however, the fluid is compressible like a gas, then equation (29) 

becomes 

a'v'd' = a"v"d" (30) 

where a' is the sectional area of the steady stream of gas at one place, if is the 
average velocity of the stream at that place, d' is the density of the gas at that 
place, and a", if' and d" are the cross-sectional area, the velocity of the stream 
and the density of the gas at another part of the stream. 

64. The ideal frictionless incompressible fluid. — ^When a jet 
of water issues from a tank, there is a certain relation between the 
velocity of the jet and the difference in pressure inside and out- 
side of the tank. When there are variations of the velocity 
of flow of water along a pipe due to enlargements or contrac- 
tions of the pipe [see equation (29)], the pressure decreases 
wherever the velocity increases and vice versa. These mutually 
dependent changes of velocity and pressure are always compli- 
cated by friction, and by the variations of the density of the fluid 
due to the variations of pressure ; and in order to gain the simplest 



HYDRAULICS. 



93 



possible idea of these mutually dependent changes of velocity 
and pressure the conception of the frictionless incompressible fluid 
is very useful. 

When the water in a pail is set in motion by stirring, it soon 
comes to rest when left to itself. A fluid which would continue 
to move indefinitely after stirring woidd he called a frictionless fluid. 

When a moving fluid is brought to rest by friction, the kinetic 
energy of the moving fluid is converted into heat and lost. Such 
a loss of energy would not take place in a frictionless fluid, 
and therefore the total energy (kinetic energy plus potential 
energy) of a frictionless fluid would be constant. This prin- 
ciple of the constancy of total energy is the basis of the following 
discussion of the flow of the ideal frictionless fluid. Indeed the 
following discussion in Arts. 65 to 67 applies not only to friction- 
less fluids but to fluids which are assumed to be incompressible 
also. 

65. Energy of a liquid. — (a) Pote^itial energy per unit of volume. 
Work must be done to pump a liquid into a region under pres- 
sure, the amount of work done in pumping one unit of volume 
of the liquid is the potential energy per unit of volume of the 
liquid in the high pressure region, and it is equal to the pressure. 
That is 

W -^p (31) 

in which W is the potential energy in ergs (or foot-poundals) of 
one cubic centimeter (or one cubic foot) of incompressible liquid 
in a region where the pressure is p dynes per square centimeter 
(or poundals per square foot). 

Proof of equation {31). Let CC, Fig. 74, be the cylinder of 
a pump which is used to pump liquid into a tank where the pres- 
sure is p, and let a be the area of the pump piston. Then ap 
is the force required to move the piston (ignoring friction), and 
apd is the work done in pushing the piston over a distance d. 
But ad is the volume of liquid pushed into the tank by the move- 
ment of the piston, and, therefore, dividing apd by the volume 
ad gives the work done per unit volume which is p. 



94 



MECHANICS. 



(6) Kinetic energy per unit volume. Let v be the velocity of a 
moving liquid and let d be the mass of unit volume (the density) 
of the liquid. Then the kinetic energy of unit volume of the 
liquid, according to equation (22) of Art. 46, is 



W' = Uv"- 



(32) 










Fig. 74. 



66. Efflux of liquid from an orifice. — A tank contains a liquid 

of density d at uniform 
pressure p. The liquid 
flows out of the tank 
through an orifice into 
the outside air where the 
pressure is ^', and it is 
required to determine the 
velocity v of the liquid in 
the jet. 

In the tank, where the 
velocity of the water is 
inappreciable, the total energy of the liquid per unit volume is 
its potential energy p as given by equation (31). 

In the jet, where the velocity is v and the pressure Is p\ the 
total energy of the liquid per unit volume is p' + ^dv^ according 
to equations (31) and (32). 

Assuming the liquid to be frictlonless the total energy of the 
water per unit volume cannot change as a given portion of water 
comes from the tank into the jet, and therefore we must have 



whence 



p = p' + ^dv^ 



d 



(33) 



Torricelli's theorem. — ^The effects of gravity are ignored in 
the above discussion, that Is to say, we have simply a tank con- 
taining water at pressure p with an orifice in it. Suppose, 
however, that the pressure p in the tank at the level of the 



HYDRAULICS. 



95 



orifice is due to gravity as in Fig. 75, or, to be more precise, the pres- 
sure-difference P — p' 
is due to the column of 
liquid of which the depth 
is X. Then p — p' = xdg, 
according to equation 
(26) of Art. 53, where g 
is the acceleration of 
gravity. Substituting 



this value of p — p' in 
equation (33) we get 




i^-^lQ 



Fig. 75. 



= ^2gX 



(34) 



and this is the velocity that would be gained by a body falling 
freely through the distance x. The relation expressed by equa- 
tion (34) was discovered by Torricelli and it is called Torricelli's 
theorem. 

67. Bernoulli's principle. — A general expression may be de- 
rived showing the mutual relationship between pressure and 
velocity along a horizontal stream of incompressible frictionless 
fluid as follows: Figure 76 represents a pipe with water flowing 




Fig. 76. 

through it. Let p' be the pressure of the water at A and let 
v' be the velocity of the water at A ; also let p" be the pressure 
of the water at B and v" the velocity of the water at B. The 
total energy per Uiiit volume of the water at A v^ p' -\- \dv''^ 
and the total energy per unit volume of the water at B is 
p'' + \dv"'^ according to equations (31) and (32). If the water 
were frictionless a given portion of the water would neither gain 



96 MECHANICS. 

nor lose energy in passing from A to B, and therefore we must 
have 

p' + W' = P" + ¥v"' (35) 

This equation shows that p'^ must be less than p' if v'^ is 
greater than v', that is, in a horizontal stream of water (assumed 
to be frictionless) the pressure must be small where the velocity 
is great and vice versa. This relation was discovered by John 
Bernoulli, and it is known as Bernoulli's principle. 

Remark. — The velocity v'' in the particular case snown in 
Fig. 76 must be greater than v\ because the sectional area of 
the pipe is smaller at B than at A. Therefore the pressure 
in the throat (at B) is less than the pressure at A. By 
using equations (29) and (35) we may easily get an expression 
for {p' — p") in terms of v' , a\ a" and d, where a' and a" 
are the sectional areas of the pipe at A and B respectively, 
and d is the density of the liquid. 

Limitations of Bernoulli's principle. — Bernoulli's principle as 
expressed by equation (35) applies only to frictionless incom- 
pressible fluids but it is approximately true for ordinary liquids 
and gases when friction is not excessive and where changes of 
pressure (and consequent changes of density) are not great. 

Bernoulli's principle, furthermore, is limited to permanent 
states of flow. 

Also Bernoulli's principle applies only to what is called irro- 
tational flow . Consider, for example, a liquid in a rapidly rotating 
bowl. The centrifugal action causes a great pressure near the 
outer walls of the bowl and it is here that the velocity is greatest, 
so that, evidently, Bernoulli's principle does not apply. 

Examples of Bernoulli's principle, (a) The disk paraaox. — 
Figures jya and 77^ represent a short piece of brass tube TT 
with a flat brass disk DD fixed to its end, and dd is a light metal 
disk. When one blows through the tube TT, the disk dd is not 
blown away from DD, but the outside air pushes dd very strongly 
against DD because of the low pressure of the rapidly moving air 
between the two disks. 



HYDRAULICS. 



97 



Figure 'j'jh shows a top view of the arrangement, and the 
small arrows in Fig. 776 represent the air blowing out from be- 
tween the edges of the disks. Consider the air stream between 
the disks, its sectional area increases towards the edge of the disks. 
In fact c'h is the sectional area of the stream at the dotted circle 



D 



y^' 



K 



D 



>A 




Fig. 77a. 
Side view. 



Fig. 77&. 
Top view. 



c' and c"h is the sectional area of the stream at the dotted circle 
c" , h being the distance between the disks (a constant) and c' 
and c" being the circumferences of the respective dotted circles. 
Evidently c"h is larger than c'h. Therefore the velocity of the 
air stream decreases towards the edges of the disks, according to 
Art. 63, and consequently the pressure of the air between the disks 
increases towards the edges of the disks. 

But the pressure of the air at the edges of the disks is atmos- 
pheric pressure. Therefore the pressure of the air between the 
disks Is everywhere less than atmospheric pressure, so that the 
outside air pushes the disks together. 

The pressure of the air in the tube TT Is not considered. 

(b) The jet pump. — The pressure in the throat B In Fig. 76 
is less than the pressure at A ; indeed the pressure at -B may be 
srnall enough to suck liquid Into the throat through the side tube 
P. Liquid thus sucked Into the throat Is carried along with the 
main stream in the pipe. Such an arrangement is called a jet 
pump. 
8 



98 



MECHANICS. 



(c) Ship suction. — Two ships steaming along side by side are 
drawn together by the action of the water. In this case a given 
particle of the water (as indicated by a small float) is stationary 
when the boats are far distant, it moves slightly as the boats 
pass by, and then comes to rest again. The water in the neigh- 
borhood of the two moving boats is not in a permanent state of flow, 
so that, as it would seem, Bernoulli's principle cannot be applied. 
But the force action between the water and the boats depends 
only on the velocity of the boats relative to the water, or on the 
velocity of the water relative to the boats. Therefore we may 
think of the boats as standing still with the water flowing steadily 
past them as indicated by the stream lines in Fig. 78. From this 




Fig. 78. 

point of view the water is In a permanent state of flow and 
Bernoulli's principle may be applied. The stream lines are 
greatly crowded together in the region between the boats, and 
only slightly crowded together in the regions on the outer sides 
of the two boats. Therefore the velocity .of the water is greater* 
between the boats than on the outer sides of the boats. There- 
fore the water levelf is lower between the boats than on the outer 
sides of the two boats, and therefore the higher level water on the 
outer sides pushes the boats together. 

(d) The curved flight of a spinning baseball. — In order to'be 

* According to equation (29) of Art. 63. 

t Change of level corresponds to change of pressure. 



HYDRAULICS. 



99 



able to apply Bernoulli's principle to the air in the neighborhood 
of a moving ball we must think of the ball as standing still (spin- 




Fig. 79. 

Blast of air blowing past a ball which is not spinning. 

ning or not as the case may be) with the air blowing past it. The 
curved lines in Fig. 79 represent a stream of air blowing past a 
ball which is not spinning, and the fine 
circles in Fig. 80 represent the motion 
of the air in the neighborhood of a 
spinning ball, the air being still except 
for the effect of the spinning ball. 
When a blast of air blows past a spin- 
ning ball as shown in Fig. 81 the 
actual velocity of the air at a is the 
sum of the velocities at a in Figs. 79 
and 80; and the actual velocity of 
the air at h in Fig. 81 is the differ- 
ence of the velocities at a in Figs. 79 
and 80. This is evident when we 
consider that the motion of the air in Fig. 81 is due to the acting 
cause of air motion in Fig. 79 and the acting cause of air motion 
in Fig. 80 together. Therefore the velocity of the air is great at 
a and small at h in Fig. 81, and therefore, according to Ber- 
noulli's principle, the pressure of the air is great at h and small 
at a, and consequently the air underneath the ball in Fig. 81 




Fig. 80. 

Whirl of air in the neighborhood 
of a spinning ball. 



100 



MECHANICS. 



pushes upwards on the ball with a greater force than the air above 
the ball pushes downwards on it. These two forces are repre- 
sented by F and/ in Fig. 82, 




/ 



Fig. 81. 
Blast of air blowing past a spinning ball. 

The force action of the air on the ball in Fig. 81 is the same 
as that which would exist if the spinning ball were moving for- 
wards through still air as indicated in Fig. 82; and the greater 
force F causes the hall to he deflected^ as indicated hy the curved 

arrow, from the direction in 
which it would continue to 
y travel if it were not spinning. 

Let us call the foremost 
' point, N, of the ball the 
noseoi the ball. It is evi- 
dent that the spinning mo- 
tion of the ball in Fig. 82 
causes the nose of the ball 
to move upwards (towards 
the top of the page) and 
the ball in Fig. 82 is deflected upwards. A spinning hall is 
always deflected -in the direction in which its nose moves. 

Bernoulli's principle is true only when friction is negligible and when the fluid 
motion is irrotational. But the air motion in Fig. 80 is due to friction between 
the air and the spinning ball, and there is only one particular case in which the 




direction of travel 



Fig. 82. 



HYDRAULICS. 



lOI 



air whirl in Fig. 80 is irrotational, namely, when the velocity of the air at a point 
is inversely proportional to the distance of the point from the center of the spin- 
ning ball. Therefore the application of Bernoulli's principle to the air motion in 
Fig. 81 is very questionable. 

The Pitot meter. — Consider a stream of water flowing past a 
pointed tube as shown in Fig. 83. The total energy per unit 
volume of the moving water is ^ + ^dv^ and the total energy 
per unit volume of the still water in the tube is p' \ therefore, 
according to Bernoulli's principle,* we have 



whence 



p + iJz;2 ^ p 



. \' d 



(i) 
(36) 



Consider a stream of water flowing sidewise past a pointed 
tube as shown in Fig. 84. In this case Bernoulli's principle does 




^ V 




Fig. 83. 



Fig. 84. 



not apply to the motion of the water at the tip of the tube, 
because the still water in the tube is separated from the moving 
water by a vortex sheet (a region of rotational motion). In fact 
the pressure of the water in the tube in Fig. 84 is the same as 
the pressure of the moving water. Similarly, the pressure of the 
fluid is the same on the two sides of the vortex sheet bb in 
Fig. 72. 

Therefore if two pointed tubes are arranged as in Figs. 83 
and 84 with their tips near together in a stream of water, the 
pressure in one tube will exceed the pressure in the other by the 

* There is some doubt as to the applicability of Bernoulli's principle to the 
liquid in the neighborhood of the tip of the tube in Fig. 83 but experiment justifies 
it because equation (36) is verified (approximately) by experiment. 



102 



MECHANICS. 



amount {p' — p) which is equal to ^dv^ according to equation 
(i). Therefore if {p' — p) is measured and the density d of 
the Hquid known, the velocity v of the stream can be calculated 
by equation (36). A pair of tubes arranged in this way with a 
device for measuring {p' — p) is called a Pilot meter from its 
inventor. 

Figure 85 shows a Pitot meter arranged for measuring the 
velocity of the water in a river. The pressure difference 
(P' — p) of equation (36) is measured by the difference of level 
h, in fact p' — p = hdg, according to equation (26) of Art. 53. 

Figure 86 shows a Pitot meter arranged for measuring the 



pipe 





\h 
i.... 



Fig. 85. 



Fie. 86. 



velocity of an air blast in a pipe. The pressure difference 
p' — p oi equation (36) is equal to hd'g where d' is the density 
of the liquid in the tube in Fig. 86. Of course d in equation 
(36) is in this case the density of the air in the blast. 

Remark. — For comparatively small changes of pressure 
Bernoulli's principle may be applied to air or gas without great 
error. 

68. Fluid friction.* — There are two fairly distinct kinds of 
friction which cause a loss of pressure as a fluid flows through a 
pipe or channel, and, although these two kinds of friction always 
exist together, there are two extreme cases in which each exists 
by itself, or nearly so. 

* No attempt is here made to discuss the fluid friction which opposes the 
motion of a boat. 



HYDRAULICS. IO3 

Viscous friction. — When a fluid flows through a very small 
smooth-bore pipe, like a small glass tube, the friction is due 
almost wholly to the sliding of each layer of fluid over the adjoining 
layer, and the volume of fluid flowing per second through such a tube 
is proportional to the pr.essure-difl^erence between the ends of the tube. 

Eddy friction. — When a fluid flows through a large pipe or 
channel, the friction is due almost wholly to the formation of 
eddies. In this case the volume of fluid flowing per second through 
the pipe is approximately proportional to the square root of the 
pressure-difl^erence between the ends of the pipe. 

69. Practical formula for calculating the frictional loss of 
pressure due to the flow of water or gas through a pipe. — The 
formula which is used in practice for calculating the frictional 
loss of pressure in a pipe is only approximately true and therefore 
the formula has no rigorous derivation. The only thing to be 
done in connection with it is to exhibit its meaning clearly, which 
is the purpose of the following argument. The flow of a fluid 
over a surface, such as the interior walls of a pipe, is opposed by a 
force which is approximately proportional to the area of the 
surface, to the density of the fluid and to the square of the velocity 
at which the fluid is flowing. Therefore, we may write 

F = kadv^ (i) 

in which a is the area of the surface, d is the density of the 
fluid, V is the velocity of flow, and F is the opposing force. 
The quantity k Is sometimes called the coefficient of friction of 
the moving fluid against the walls of the pipe. It depends 
greatly upon the degree of roughness of the walls. 

Consider a pipe of which the length is L and of which the 
inside diameter Is D. The total area of Interior walls of this 
pipe is TzDL, so that, using ttDL for a in equation (I), we have 
F = kirDLdv^ for the total opposing force acting on a fluid of 
density d which flows through the pipe at velocity v. This 
opposing force is equal to the difference of pressure at the two 



ends of the pipe multiplied by the sectional area I 1 of the 



(t) 



104 MECHANICS. 

bore of the pipe. Therefore, using p for the loss of pressure aue 
to friction, we have 

F = P = kTvDLdv^ (ii) 

whence 

p = ^^-^r (37) 

It is usually convenient to express the loss of pressure in terms 

of the volume V of fluid discharged per second instead of 

expressing it in terms of the velocity v of the fluid. According 

TT /\.V 

to Art. 63 F = wr^v = - -D^v, so that v = —^. Therefore, sub- 

4 tV"^ 

stituting this value for v in equation (37), we get 

e/LkLdV^ , , 

P = —.^ (38) 

If the loss of pressure p is expressed in '' pounds " per square 
foot, the length of the pipe L in feet, the diameter D of the 
pipe in feet, the density d of the fluid in pounds per cubic foot, 
and the discharge rate V in cubic feet per second, then the value 
of k is about 0.000082 for water in ordinary cast iron pipes and 
about 0.0000557 for gas or air in cast iron pipes. 

Example. — It is required to find the inside diameter D oi a 
cast iron pipe to bring 10 cubic feet of water per second from a 
reservoir to a distributing point in a city, the length of the pipe 
being 16,000 feet and the allowable loss of pressure being that 
which corresponds to a head of 200 feet of water. That is, if 
the reservoir is 350 feet above the distributing point, then there 
is to be an available head of 150 feet at the distributing point. 

According to equation {2^) of Art. 53 the pressure correspond- 
ing to 200 feet head of water is 200 X 62 J "pounds" per square 
foot. Therefore, substituting in equation (38) k = 0.000082, 
L = 16,000 feet, d = 62J pounds per cubic foot, F = 10 cubic 
feet per second, and p = 12,500 "pounds" per square foot, 
we get D = 1 .336 feet as the required diameter of pipe. 



PART II. 



THE THEORY OF HEAT. 



A very interesting book is Tyndall's Heat, A Mode of Motion, published in 1875; 
sixth edition, revised, in 1880. 

Good books for the beginner are Poynting and Thomson's Heat (Griffin & Co., 
1902) and Edser's Heat for Advanced Students (Macmillan & Co., London, 1899). 

Books on thermodynamics and on the atomic theory. 

Thermodynamik, Max Planck, Leipsig, 1905. This book has" been translated 

into English and into French. 
Thermodynamics, Edgar Buckingham, The Macmillan Co., 1900. 
Kinetic Theory of Gases, O. E. Meyer, translated by R. E. Baynes, Longmans 

Green & Co., London, 1899. 
Vorlesungen iiber Gas Theorie, L. Boltzmann, Leipzig, 1896. 

The student of the atomic theory must be familiar with the theory of prob- 
ability. A good modern book on this subject is The Mathematical Theory of Prob- 
abilities by A. Fisher, translated by W. Bonynge, The Macmillan Co., New York, 
1915- 
Books applying thermodynamics and the atomic theory to chemistry. 

Theoretical Chemistry, W. Nernst, translated by R. A. Lehfeldt; new edition 

revised by H. T. Tizard, Macmillan & Co., 1916. 
An Introduction to the Principles of Physical Chemistry, E. W. Washburn, 

McGraw-Hill Book Co., New York, 191 5. 
The Phase Rule and its Applications, AlexQ.nder Findlay, Longmans Green & 

Co., 1911. 
Theory of Solution and Electrolysis, W. C. D. Whetham, Cambridge, 1902. 

Radiation as a branch of the theory of heat. 

Vorlesungen iiber die Theorie der Wdrmestrahlung, Max Planck, Leipsig, 1913. 
A good discussion of this, as of nearly every topic in physics, is given in Winkel- 

mann's Handbuch der Physik. 
A very good discussion of Optical Pyrometry is given by Waidner and Burgess, 

Bulletin of the Bureau of Standards, Vol, I, pages 189-254, February, 1905. 

Heat conduction. 

The classical book by Fourier is entitled The Theory of Heat; an important 
book in this connection is Fourier's Series and Spherical Harmonics, W. E. 
Byerly, Ginn & Co., 1895. 
Statistical Physics. 

There are four distinct methods in physics, namely, the method of Mechauics, 
the method of Thermodynamics, the method of the Atomic Theory, and 
the Statistical Method. Thermodynamics and the Atomic Theory are 
discussed and contrasted in Article 79 of this treatise; Mechanics and the 
Atomic Theory are discussed and contrasted in article 227. The use of the 
Statistical Method in physics is discussed in a very simple paper by W. S. 
Franklin, Science, Vol. XLIV, pages 158-163, August 4, 1916. 

105 



CHAPTER VI. 

TEMPERATURE AND THERMAL EXPANSION. 

70. Thermal equilibrium. Temperature. — ^When a substance 
(or a system of substances) is left to itself and shielded from all 
outside distributing influences it settles to a quiet state or condi- 
tion in which there is no tendency to further change of any kind. 
This quiet state is called a state of thermal equilibrium. For 
example, the various objects in a closed room or cellar settle to 
thermal equilibrium; when a piece of red-hot iron is thrown into 
a pail of water, the mixture, at first turbulent, becomes more 
and more quiet and finally reaches a state of thermal equilibrium. 

A number of bodies which have settled to a common state of 
thermal equilibrium are said to have the same temperature. 
Thus a number of bodies left in a closed cellar have the same 
temperature. 

Although the various objects in a closed cellar are at the same 
temperature, some of the objects may feel cooler than the others 
to the hand. In fact a cold piece of metal feels cooler than a 
piece of dry wood at the same temperature. Likewise a piece 
of metal taken out of an oven feels hotter than a piece of wood 
taken out of the same oven. 

71. Coexistent phases. — The forms of a given substance which 
can exist together in thermal equilibrium are called coexistent 
phases of that substance. Thus the water in a vessel may be 
partly liquid water and partly ice (a liquid phase and a solid 
phase). The water in a steam boiler is partly liquid water and 
partly steam (a liquid phase and a vapor phase). 

Phases of the same composition. — ^Water may be wholly 
converted into ice, and ice may be wholly converted into water. 
Therefore water and ice are said to be phases of the same com- 
position. 

107 



I08 THE THEORY OF HEAT. 

Phases of different composition. — ^When a hot solution of sugar 
is allowed to cool it separates into two phases, namely, sugar 
crystals and a remnant of weakened solution or syrup. These 
are called phases of different composition because sugar crystals 
cannot alone be converted into syrup, and syrup cannot be 
wholly converted into sugar. 

This breaking up of solutions into phases of different composi- 
tion is the fundamental fact of chemistry. Thus when solutions 
of sodium chloride and silver nitrate are mixed, the mixture settles 
to thermal equilibrium with a solid phase consisting of precipi- 
tated silver chloride and a liquid phase consisting of a solution of 
sodium nitrate. When a solution of salt is evaporated more and 
more we have eventually a vapor phase consisting of pure water 
vapor (which we have allowed to escape) and a residual solid 
phase of pure salt. 

Elementary substances and compound substances. — A salt 
solution is called a compound substance because it can be broken 
up into phases of different composition, namely, water and salt. 
The component parts of a compound substance may themselves 
be compound. Thus water may be separated into phases of 
different composition, namely, hydrogen gas and oxygen gas, 
by means of the electric current. Substances which have never 
yet been broken up into phases of different composition are 
called elementary substances or chemical elements. For example, 
oxygen, hydrogen, iron, lead, sulphur, etc., are chemical elements. 

Chemical compounds and mixtures. — The component parts of 
some compound substances always occur in unalterably fixed 
proportions ; such compound substances are called chemical com- 
pounds. On the other hand, there are many compound sub- 
stances of which the component parts may be varied at will 
between wide limits; such compound substances are called 
mixtures. Two substances are said to combine when they form 
a chemical compound, and they are said to mix when they form 
a mixture. There is, however, no sharp line of distinction 
between chemical compounds and mixtures. 



TEMPERATURE AND THERMAL EXPANSION. I09 

72. Decrease of volume of a gas with increase of pressure. 
Boyle's law. — Solids and liquids, generally, decrease but slightly 
in volume when subjected to increase of pressure. Thus the 
volume of water decreases about one part in twenty thousand 
when its pressure is increased from one atmosphere to two 
atmospheres. The volume of steel decreases about one part in 
a million when the pressure is increased from one atmosphere to 
two atmospheres. 

Gases, on the other hand, decrease greatly in volume when 
subjected to increase of pressure. The remarkable contrast 
between water and air in regard to compressibility may be 
shown by filling a bicycle pump with air and with water, and 
striking the piston rod in each case with a small hammer. The 
air will be found to behave like a soft cushion, and with the water 
it will seem as if the barrel and piston and piston rod were one 
solid piece of steel. 

When the temperature of a gas is kept constant, the volume of the 
gas is inversely proportional to the pressure to which the gas is 
subjected. That is 

k 

P 
or 

pv = k (39) 

in which v is the volume of a given amount of gas, p is the 
pressure of the gas, and ^ is a proportionality constant. This 
relation, known as Boyle's law, was discovered by Robert Boyle* 
in 1660, and more completely established by Mariotte, who 
discovered it independently in 1676. Boyle's law is very nearly 
exact for such gases as hydrogen, nitrogen and oxygen at ordinary 
temperatures and moderate pressures, but all gases deviate more 
or less from the law, especially at very low temperatures and at 
very high pressures. 

73. Thermal expansion of gases. Gay Lussac*s law. — All 
gases expand equally for the same rise of temperature. The exact 

* New Experiments touching the Spring of Air, Oxford, 1660. 



no 



THE THEORY OF HEAT. 



meaning of this statement may be made clear as follows: A 
number of cylinders contain equal volumes of gases in a cellar 
as shown in Fig. 87a, and the cylinders are carried to a warm 
room (given rise of temperature). Then, when the volume of each 
gas is allowed to increase so that its pressure may he the same as it 
was in the cellar, the increase of volume is the same for all, as 
indicated by the changed positions of the pistons in Fig. 87^. 



in cool cellar 



in warm room 





III" '""IIH 



nitrogen 



Fig. 87a. 



Fig. 876. 



Following are two precise statements of Gay Lussac's law: 

(a) When equal volumes of various gases are heated they all 
expand equally (for the same rise of temperature) if their pres- 
sures are kept constant. 

(6) When various gases at the same initial pressure are heated 
and not allowed to expand the pressure increase is the same for 
all for the same rise of temperature. 

74. The measurement of temperature. — To measure a thing 
is to divide it into equal parts and count the parts, as so many 
feet of length, or so many seconds of time.* One cannot divide 

* In many kinds of measurement the two distinct operations, (c) dividing into 
equal unit parts, and (&) counting of parts, are obscured by the use of more or less 
elaborate measuring devices, but every measurement does in fact consist of these 
two fundamental operations. In measuring a length by means of a scale of inches 
the operation of dividing into equal parts has been performed once for all by the 
maker of the scale, and the counting is also "ready-made" by the numbers stamped 
on the scale. In weighing a consignment of coal the operation of dividing into 
equal parts has been performed once for all by the maker of the set of weights, and 
the counting is also "ready-made" by the numbers stamped on the weights and on 
the balance beam. 



TEMPERATURE AND THERMAL EXPANSION. Ill 

a temperature into equal parts, and therefore in a certain funda- 
mental sense temperatures cannot be measured. Indeed tem- 
perature can be measured only in terms of some temperature 
effect which is measurable, and the most obvious temperature 
effect for this purpose is expansion. 

There is, however, an objection to the use of expansion for 
the measurement of temperature because every substance has 
its own characteristic irregularities of expansion so that there is 
an element of extreme arbitrariness in the choice of a particular 
substance in terms of whose expansion we agree to measure 
temperature. But, according to Gay Lussac's law, all gases 
expand alike, and therefore it would seem to be best to measure 
temperature by the thermal expansion of a gas, or by the increase 
of pressure* of a gas with increase of temperature. 

Air-thermometer temperatures and the air thermometer. — Let 
us agree to consider one temperature T' as twice as great as 
another temperature T when the pressure of a constant volume 
of air is twice as great at temperature T' as at temperature T. 
This agreement can be expressed generally by the equation 

in which a constant volume of gas has pressure p at temperature 
T and pressure p' at temperature T\ 

The air thermometer is an arrangement for measuring the ratio 
of two temperatures in accordance with equation (40). A glass 
or porcelain bulb contains a fixed quantity of dry air, and a small 
tube connects the bulb to a pressure measuring dev^ice.f Of 
course the bulb expands slightly with rise of temperature and 
the air in the connecting tube is not heated with the air in the 
bulb, but for the sake of simplicity let us assume that the volume 
of the bulb is constant and that the volume of the connecting 
tube is negligible. t Then to measure the ratio of two tempera- 

* Volume being constant, 
t See Arts. 54 and 55. 

t These two complications are always taken into account in precise air-ther- 
mometer measurements. 



112 THE THEORY OF HEAT. 

tures we place the bulb in the region where the temperature is T 
and measure the pressure p of the air in the bulb, then we place 
the bulb in the region where the temperature is T' and measure 
the pressure p' of the air in the bulb. Then T'jT = p'jp 
according to equation (40). 

Example showing the use of the air thermometer. — The bulb 
of an air thermometer is placed in a steam bath at temperature 
5, and the pressure p' of the air in the bulb is measured ; the 
bulb is then placed in ice water at temperature /, and the 
pressure p of the air in the bulb is measured. The ratio p'jp 
is found to be equal to 1.367, and, therefore, according to 
equation (40) we have 

-j= 1-367 (41) 

• 

The steam bath is supposed to be pure steam (without admixture 
of air) in the presence of condensed droplets of water, and the 
pressure of the steam is supposed to be standard atmospheric 
pressure, namely, 760 millimeters of mercury.* The temperature 
5 under these conditions is called standard steam temperature; 
and the temperature I is called ice temperature.] 

Further data needed before the value of any given tempera- 
ture can be measured. — It is evident from the above discussion 
that the air thermometer measures only the ratio of two tem- 
peratures. To he able to measure the value of any given temperature 
with the air thermometer it is necessary to assign an arbitrary 
value to some fixed reference temperature such as standard steam 
temperature 5 or ice temperature 7. Instead of doing this, 
however, an arbitrary value of 100 degrees has been chosen for 
the difference {S — T). That is, by agreement, we have 

5 - J = 100° (42) 

* See Art. 54. 

t The ice water is supposed to be pure. The pressure of the ice water need 
not be specified because the ordinary variations of atmospheric pressure produce 
only imperceptible variations of the melting temperature of ice. 



TEMPERATURE AND THERMAL EXPANSION. 113 

Therefore, solving equations (41) and (42) we get 

/ = 273° (43) 

S = 373° (44) 

Having thus determined the values of I and S (partly by 
choice, of course), the value of any temperature can be deter- 
mined by measuring its ratio to / or to 5* by means of the air 
thermometer. Temperature values determined in this way are 
called Kelvin temperatures'^ to distinguish them from tempera- 
tures reckoned upwards from the ice point. Thus I = 2'J2>°. K. 
and 5* = 373° K., and the temperature of melting lead, for 
example, is 6oo° K. 

75. Formulation of Gay Lussac^s law and Boyle's law. — When 
temperatures are measured by the air thermometer as above 
explained then as a matter of course the pressure of a constant 
volume of air is proportional to its Kelvin temperature; and Gay 
Lussac's law reduces merely to the substitution of the words any 
gas for the word air in this statement. 

Representing Kelvin temperature by T, Boyle's law and 
Gay Lussac's law may both be formulated thus : 

pv = R'T (45) 

in which p is the pressure and v is the volume of a given amount 
of gas at any Kelvin temperature T, and R' is a proportionality 
factor the value of which depends on the amount of the gas in 
grams. A more general form of this equation is 

pv = MRT (46) 

in which M is the mass of the gas in grams and i? is a constant 
which depends only on the molecular weigh tf of the gas. 

76. The mercury-in-glass thermometer.l — The most con- 

* Kelvin temperature is often called absolute temperature. 

t Thus the molecular weight of oxygen is 32, of nitrogen is 28, of hydrogen is 2, 
of water vapor is 18, and so on. 

% A good description of the construction of a mercury-in-glass thermometer is 
given on pages 4-14, and special forms of thermometer for indicating maximum 

9 



114 THE THEORY OF HEAT. 

venient device for measuring temperature is the ordinary 

mercury-in-glass thermometer with which every one is famihar. 

A glass tube AB, Fig. 88, of fine uniform bore, with a bulb at 
one end, is filled with mercury at a temperature some- 
^ what above the steam point and the tube is sealed at 
A. As the instrument cools, the mercury contracts 
more rapidly than the glass and thus only partly fills the 
stem. The instrument is then placed in an ice bath and 
the position of the surface of the mercury in the stem is 
marked at /. Then the instrument is placed in a steam 
bath at standard atmospheric pressure, and the steam 
point is marked at 5. 

In the centigrade scale {Celsius),'^ which is the scale 
universally used in scientific work, the distance SI is 
divided into loo equal parts, which divisions are con- 
tinued above 5 and below /. These marks are 
numbered upwards beginning at I which is number 

' iiii zero. The marks below / are numbered negatively 

■ from /. 
Any temperature is specified by giving the number of 
the mark at which the mercury stands when the ther- 
mometer is brought to that temperature. For example, 

Fig. 88. 

65° C. (read sixty-five degrees centigrade) is the tempera- 
ture at which the mercury in a mercury-in-glass thermometer 
stands at mark number 65 of the centigrade scale. 

temperatures and minimum temperatures are described on pages 18-20 of Edser's 
Heat for Advanced Students, Macmillan & Company, London, 1908. 

A device for measuring very high temperatures is called a pyrometer. A good 
discussion of the older methods for measuring high temperatures is given in Bur- 
gess's translation of High Temperature Measurements by Le Chatlier and Boudou- 
ard, John Wiley & Sons, New York, 1904. An excellent discussion of optical 
methods for measuring high temperatures is given by Waidner and Burgess, Bulletin 
of the Bureau of Standards, Vol. I, pages 189-254, February, 1905. 

* The only other thermometer scale of which mention need be made is that of 
Fahrenheit in which the distance SI is divided into 180 equal parts, which divisions 
are continued above 5 and below I. These marks are numbered upwards begin- 
ning with the thirty-second mark below I which is number zero. The marks below 
zero are numbered negatively. 



TEMPERATURE AND THERMAL EXPANSION. 



115 



Mercury-in-glass temperatures. — The indications of an accu- 
rately constructed mercury-in-glass thermometer are slightly 
different from air- thermometer temperatures (reckoned from ice 
point) because of the irregularities in the expansion of mercury 
and glass, and temperature values as indicated by an accurate 
mercury-in-glass thermometer made of a standard variety of 
glass are called mercury-in-glass temperatures. 

The following table shows air-thermometer temperatures (reckoned from ice 
point) and mercury-in-glass temperatures (Jena normal glass) corresponding to 
hydrogen-thermometer temperatures reckoned from the ice point. All three 
thermometers agree, of course, at ice point and at steam point, and the differences 
for the intervening temperatures depend upon irregularities of expansion. Thus, 
the difference between the hydrogen-thermometer temperatures and the air- 
thermometer temperatures shows that these gases do not both expand in exactly 
the same way with rise of temperature, and another difference between the hydrogen 
and the air thermometers which does not appear in the table is that the ratio of 
steam temperature to ice temperature as measured by the hydrogen thermometer 
is slightly different from the ratio as measured by the air thermometer. 





TABLE.* 




Comparison of Hydrogen, Air and Mercury-in- 


Glass Temperatures. 


Hydrogen-thermometer 

temperatures (reckoned 

from ice point) 


Air-thermometer 

temperatures (reckoned 

from ice point) 


Mercury-in-glass 

temperatures (Jena 

Normal Glass) 


n-273. 


ra-273. 


C. 


0.° 


0.° 


0° 


10. 


10.007 


10.056 


20. 


20.008 


20.091 


30. 


30.006 


30.109 


40. 
50. 
60. 
70. 
80. 
90. 


40.001 
49-996 
59-990 
69.986 

79-987 
89.990 


40.11 1 
50.103 
60.086 
70.064 
80.041 
90.018 


100. 


100. 


100. 



Remark. — For most purposes the variations in this table are 
negligible. A temperature measured by an air or hydrogen ther- 
mometer as explained in Art. 74 is considered to be the true Kelvin 
temperature {see Art. no), and the Kelvin temperature is found 
from ordinary centigrade mercury-in-glass temperatures by 
adding 273 to C. 

* From Landolt and Bornstein's Physikalisch-Chemische Tabellen, page 93. 



Il6 THE THEORY OF HEAT. 

Standard thermometers. — It is of course impossible to con- 
struct a thermometer so that the bore of the stem is perfectly 
uniform, and slight errors are always made in the location of the 
ice and steam points and in the marking of the divisions on the 
stem. A standard thermometer is a thermometer of which the 
errors have been determined* so that the true mercury-in-glass 
temperature corresponding to any given reading is known. No 
thermometer which has not been standardized is to be depended 
upon for work of even moderate accuracy. f 

77. Thermal expansion of liquids and solids. — In general, 
liquids and solids expand with rise of temperature. This is 
illustrated by the fact that a long line of steam pipe has to be 
provided with a telescope joint to allow for expansion and con- 
traction inasmuch as the temperature of the pipe is apt to be 
changed at any time from ordinary air temperature to steam 
temperature when the steam is turned on, or from steam tem- 
perature to air temperature when the steam is turned off. The 
movement of the mercury column in the stem of a thermometer 
shows that mercury expands more rapidly than glass as the tem- 
perature rises. The expansion of the glass causes the bulb to 
grow larger but the greater expansion of the mercury causes the 
mercury to rise in the stem. 

The expansion of a gas (at constant pressure) is very much 
greater than the expansion of a liquid or solid, and all gases 
expand very nearly alike (Gay Lussac's Law), whereas every 
liquid and every solid exhibits characteristic peculiarities, expand- 
ing more rapidly at certain temperatures than at others, and in 
some cases actually contracting with rise of temperature. Most 
liquids exhibit marked irregularities of expansion near their 
freezing points. Thus, water contracts as it is heated from 0° C. 

* A good discussion of the standardization of a mercury-in-glass thermometer is 
given on pages 23-38 of Edser's Heat for Advanced Students. A discussion of the 
use of a mercury-in-glass thermometer is given on pages 140-143 of FrankHn, Craw- 
ford and MacNutt, Practical Physics, Vol. I. 

t A well-made thermometer can be sent to the United States Bureau of Stand- 
ards, Washington, D. C, where it will be standardized for a small fee. 



TEMPERATURE AND THERMAL EXPANSION. 



117 



to 4° C. at which temperature the volume of a given mass of 
water is a minimum or its density is a maximum; and beyond 
4° C. water increases in volume with rise of temperature, at first 
slowly and then more and more rapidly as the temperature rises. 
Coefficient of linear expansion of a solid. — Let Lo be the 
length of an iron bar at 0° C, and let Lt be its length at /° C. 
The increase of length from 0° C. to f C. is Lt — Lq, and this 
increase of length is accurately proportional to the initial length 
Lq of the bar and approximately proportional to the rise of 
temperature /. Therefore we may write 

Lt — Lq = aL4 (i) 

whence 

Lt = Lo(i + at) (47) 

The proportionality factor a is called the coefficient of linear 
expansion* of the iron, and, according to equation (i), it is equal 
to the increase of length per unit initial length per degree rise 
of temperature. 

Coefficient of cubic expansion of a liquid or solid. — Let Fo 
be the volume of a given substance at 0° C. and let Vt be its 
volume at f C. The increase of volume from 0° C. to /° C. is 
Vt — Fo, and it is accurately proportional to the initial volume 
Fo and in many cases approximately proportional to the rise of 
temperature /. Therefore we may write 

F, - Fo = (3Vot (ii) 

whence 

Vt = Fo(i + ^t) (48) 

The proportionality factor /? is called the coefficient of cubic 
expansion! of the substance, and, according to equation (ii), it 
is equal to the increase of volume per unit initial volume per 
degree rise of temperature. 

78. Irregularities of expansion of solids. — Solids show irregu- 
larities of expansion which are in some cases as marked as the 
irregularities of expansion of liquids near their freezing points. 

* Strictly the mean coefficient of linear expansion between 0° C. and t° C. 
t Strictly the mean coefficient of cubic expansion between 0° C. and t° C. 



Il8 THE THEORY OF HEAT. 

These irregularities occur at what are called transition tempera- 
tures, a transition temperature for a given substance being a 
temperature below which the substance is in one crystalline form 
and above which the substance is in another crystalline form. 
The most familiar example of a transition temperature is the 
so-called temperature of recalescence of steel.* 

Solids exhibit peculiarities of expansion which are not ex- 
hibited by liquid and gases. Thus, many solid substances do 
not expand promptly with rise of temperature or contract prompt- 
ly with fall of temperature, the ultimate change of dimensions 
corresponding to a given change in temperature requiring in 
some cases days or even months before it is established. The 
best known example of this tirae-lag of expansion is furnished 
by ordinary glass. The mercury column of a mercury-in-glass 
thermometer which has been kept for a long time at room tem- 
perature and which is suddenly brought to steam temperature 
rises at first too high, and as the bulb slowly expands to the 
ultimate size which corresponds to steam temperature the mer- 
cury column slowly drops to its correct position. 

A most interesting substance is the non-expansible nickel-steel 
alloy which was discovered by Guillaume, a nickel-steel contain- 
ing 36 per cent, of nickel, and known as invar. Its coefficient of 
expansion is less than one tenth of that of ordinary steel. The 
increase in length of a meter scale made of invar when it is heated 
from 0° C. to 100° C. is a little less than one tenth of a millimeter, 
whereas a meter scale made of ordinary steel would increase in 
length by about i .3 millimeters for the same rise of temperature. 
This alloy, invar, is very sluggish in its expansion and contrac- 
tion. When the increase of temperature is small the increase 
of length does not fully reach its final value in two months. 
Therefore when a bar of invar is subjected to fluctuations of 
temperature which are neither very large nor very long con- 
tinued the change of length of the bar is extremely small and for 
many purposes negligible. 

* See Art. 10 1. 



CHAPTER VIL 

THE FIRST LAW OF THERMODYNAMICS. CALORIMETRY. 

79. Atomic theory of heat and thermodynamics.* — In nearly 
every branch of physical science there are two more or less dis- 
tinct methods of attack, namely, (a) a method of attack in which 
the effort is made to develop conceptions of the physical processes 
of nature, and (&) a method of attack in which the attempt is 
made to correlate phenomena on the basis of observed data. 
In the theory of heat, the first method is represented by the 
application of the atomic theory to the study of heat phenomena, 
and the second method is represented by what is called thermo- 
dynamics. In the first case one tries to imagine the nature of 
such a process as the melting of ice or the burning of coal, and 
in the second case one is content to measure the amount of heat 
absorbed or given off and to study the physical properties of the 
substances before and after the change takes place. 

The atomic theory is used in every branch of physics, f but we 
are here concerned with the use of the atomic theory in the study 
of heat phenomena (which includes the whole of chemistry). 
Every student of elementary chemistry is to some extent familiar 
with the use of the atomic theory in the building up of clear ideas 
of chemical action, and the atomic theory of gases is a highly 
mathexnatical phase of the atomic theory of heat. 

In addition to these two highly developed branches of the 
atomic theory of heat (chemistry and the theory of gases) we 
have another important phase of the atomic theory of heat, 
namely, that phase which is best exemplified by Tyndall's 
classical book entitled Heat, a Mode of Motion. % Following is 
a quotation from this book: ''When a hammer strikes a piece 

* The term thermodynamics is here used in its proper significance, meaning the 
whole of the theory of heat except those parts which involve the atomic theory. 

t Chapter XVI, for example, is a brief discussion of what is, perhaps, the most 
important branch of atomic theory, namely, the atomic theory of electricity. 

X This book appeared about 1875; sixth edition revised in 1880. It should be 
read by every student who wishes to understand the phenomena of heat in terms 

119 



120 THE THEORY OF HEAT. 

of lead, the motion of the hammer appears to be entirely lost. 
Indeed, in the early days it was supposed that what we now call 
the energy of the hammer was destroyed. But there is no loss. 
The motion of the massive hammer is transformed into molecular 
motion in the lead, and here our imagination must help us. 
In a solid body, although the force of cohesion holds the atoms 
together, the atoms are supposed, nevertheless, to vibrate within 
certain limits. The greater the amount of mechanical action 
invested in the body by percussion, compression, or friction, the 
greater will be the rapidity and the wider the amplitude of the 
atomic oscillations. 

"The atoms or molecules thus vibrating, and, as it were, seek- 
ing wider room, xirge each other apart and cause the body of which 
they are the constituents to increase in volume. By the force of 
cohesion, then, the molecules are held together; by the force of 
heat (molecular vibration) they are pushed asunder;* and the 
relation of these two antagonistic powers determines whether 
the body is a solid, a liquid or a gas. Beginning with a solid 
substance, every added amount of heat pushes the molecules 
more widely apart; but the force of cohesion acts more and 
more feebly as the distance through which it acts is augmented. 
Therefore, as the expansive effect of heat grows strong, its op- 
ponent, cohesion, grows weak until finally the particles are so 
far loosened from each other as to be at liberty; not only to 
vibrate to and fro across a fixed position, but also to roll or glide 
around each other. Cohesion is not yet entirely destroyed, f but 
it is modified so as to permit the particles of the substance to 
glide over each other. This is the liquid condition of matter. 

of molecular motion. To attempt to develop these general ideas in an elementary 
text on heat is out of the question; an elementary text on heat must be devoted 
primarily to thermodynamics. The quotation above is not given in Tyndall's 
exact words. 

* These two statements by Tyndall are not always true. Thus when ice changes 
to water contraction takes place, and the average distance between the molecules 
decreases. 

t It is a familiar fact that the different parts of a drop of water cling together, or, 
in other words, the force of cohesion is not entirely absent in water. 



THE FIRST LAW OF THERMODYNAMICS. 



121 



" In the interior of a mass of liquid the motion of every molecule 
is limited and controlled by the molecules which surround it. 
But when sufficient heat is imparted to a liquid at a point the 
molecules break the last fetters of cohesion and fly asunder to 
form a bubble of vapor. At the free surface of a liquid it is 
easy to conceive that some of the vibrating molecules may escape 
from the liquid and wander about through space. Thus freed 
from the influence of cohesion we have matter in the gaseous 
form." 

Thermodynamics. — To understand the essential features of the 
science of thermodynamics, it is necessary to revert to the dis- 
cussion of work and energy. Whenever a substance, or a system 
of substances, gives up energy which it has in store, the substance 
or system of substances always undergoes change. Thus, the fuel 
which supplies the energy to a steam engine and the food which 
supplies the energy to a horse undergo a chemical change; the 
steam which carries the energy of the fuel from the boiler to 
the engine cools off or undergoes a change of temperature when it 
gives up its energy to the engine; and so on. 

Not only does a substance undergo a change when it gives up 
energy by doing work, but a substance which receives energy or 
has work done upon it undergoes a change. Thus, when air is 
compressed in a bicycle pump work is done on the air and the 
air becomes warm; when work is done upon a coin in rubbing it 
upon a board, the coin becomes warm; and so on. 

In Chapter III the theory of energy was discussed in connec- 
tion with mechanical changes only, thermal and chemical changes 
being carefully ignored. We are now, however, to take up the 
study of thermal and chemical changes, and it is important at 
the outset to understand two things as follows: 

(a) Our study is not to be concerned with thermal and chemical 
actions themselves, but with their results. The changes them- 
selves are, as a rule, extremely complicated. Thus, the details of 
behavior of the coal and air in a furnace are infinitely complicated. 
The important practical thing, however, is the amount of steam 



122 THE THEORY OF HEAT. 

that can be produced by a pound of coal, and this depends only 
upon (i) the condition of the water from which the steam is 
made, that is, whether the water is hot or cold to start with, (2) 
the condition of the air and of the coal which are to combine in 
the furnace, (3) the pressure and temperature of the steam which 
is to be produced, and (4) the condition of the flue gases as they 
enter the chimney. That is to say, it is necessary to consider 
only the state of things before and after the combustion takes place, 
and the only measurements that need be taken are measurements of 
substances in thermal equilibrium. 

(b) The other important thing is that in studying thermal and 
chemical changes we have to do with a new kind of energy. The 
gravitational energy of an elevated store of water can be wholly 
converted into mechanical work,* the energy of two electrically 
charged bodies can be wholly converted into mechanical work, 
the kinetic energy of a moving car can be wholly converted into 
mechanical work, and so on. On the other hand, the energy of 
the hot steam which enters a steam engine from a boiler cannot 
be wholly converted into mechanical work. 

Any store of energy which can be wholly converted into 
mechanical work may be called mechanical energy. The energy 
of the hot steam which enters a steam engine from a boiler is 
called heat energy. The important difference between mechanical 
energy and heat energy, namely, that one can be wholly converted 
into mechanical work whereas the other cannot, may be clearly 
understood in terms of the atomic theory: Every particle of a 
moving car travels in the same direction and all of the particles 
work together to produce mechanical effect when the car is 
stopped; the molecules of hot steam, however, fly to and fro 
in every direction, and no method can be devised whereby the 
whole of the energy of the erratically moving molecules of hot 
steam can be used to produce mechanical effect. 

* Any energy which is converted into heat because of friction exists in the form 
of mechanical energy or work before it is so converted, and this fact must be kept 
in mind in connection with the statements above given as to the conversion of 
gravitational and electrical energy into mechanical work. 



THE FIRST LAW OF THERMODYNAMICS. 



123 



80. The dissipation of mechanical energy.* Preliminary state- 
ment of the first law of thermodynamics. — In the attempt to 
exclude all thermal changes from the purely mechanical discussion 
of energyf we were confronted by the fact that friction (with its 
accompanying thermal changes) is always in evidence every- 
where. In every actual case of motion, the moving bodies are 
subject to friction and to collision, their energy is dissipated, 
and they come to rest. This dissipation of energy is always 
accompanied by the generation of heat.J 

It is important to understand that the term " dissipation of 
energy " refers to the conversion of mechanical energy into heat 
by friction or collision.! Thus, energy is dissipated in the bearing 
of a rotating shaft, energy is dissipated when a hammer strikes 
a nail, and so on. The atomic theory enables one to form a clear 
idea of the dissipation of energy. Thus the energy of the regular 
motion of a hammer is converted into energy of irregular || molec- 
ular motion when the hammer strikes a nail. 

* The dissipation of energy is sometimes spoken of as the degradation of energy 
from any form which is wholly available for the doing of mechanical work into heat. 

t See Arts. 40 to 48 

X One of the most important steps in the establishment of the principle of the 
conservation of energy in its general form, which includes heat energy, was made by 
Count Rumford in his experiments on the generation of heat in the operation of 
boring cannon. The results of these experiments were published in the Philosoph- 
ical Transactions for 1799. The first clear statement of the principle of the con- 
servation of energy in its general form was published in 1842 by Julius Robert 
Mayer. The celebrated experiments of Joule on the heating of water by the dis- 
sipation of work were commenced in 1840 These experiments are described on 
pages 274-278 of Edser's Heat for Advanced Students. The most accurate inves- 
tigation on the heating of water by the dissipation of mechanical energy is the 
work of Rowland in 1879. 

§ Mechanical energy is also dissipated in a wire in which an electric current is 
flowing. 

II A substance in thermal equilibrium exhibits no visible motion and therefore 
a state of thermal equilibrium has been called a quiescent state Very violent 
molecular motion is supposed, however, to exist when a substance is in thermal 
equilibrium, but this molecular motion is of the same average character in every 
part of the substance. 

It requires some power of imagination to think of a substance as being composed 
of a great number of small particles (molecules) in incessant and irregular motion, 
and to think of the energy of a moving hammer as still existing by virtue of an 



124 



THE THEORY OF HEAT. 



8i. The first law of thermo- 
dynamics. — The extension of the 
principle of the conservation of 
energy to include heat (and 
chemical) effects is called the first 
law of thermodynamics. 

A given substance A is heated 
hy the dissipation of mechanical 
work and then brought back to its 
initial condition by being brought 
into contact withanother cooler sub- 
stance B, then the change which 
is produced in substance B is 
exactly the same^ as would be pro- 
duced if substance B were to be 
heated directly by the dissipation 
of the original amount of mechan- 
ical work. 

That is to say, a substance 
which has been heated by the 
dissipation of work stores something which is equivalent to the 
work, and this something is called heat. 

increased violence of molecular motion after a hammer blow. Every student of 
physics should see the irregular and incessant to-and-fro motion of very fine particles 
suspended in water, using a good microscope. This motion was discovered by the 
English botanist, Brown, in 1827, and it is called the Brownian motion. The Brown- 
ian motion is the irregular molecular motion of the water rendered visible (and 
greatly reduced in amplitude) by the small suspended particles. 

To see the Brownian motion, grind a small amount of insoluble carmine in a few 
drops of water by rubbing with the finger in a shallow dish, place a drop of the 
mixture on a microscope slide, and use a magnifying power of about 400 diameters. 
The particles in India ink are much finer than the particles of carmine and a higher 
magnifying power is required to see them. 

A very interesting discussion of the present position of the atomic theory is 
given by Ernest W. Rutherford in his address before the Physics Section of the 
Winnipeg meeting of the British Association for the Advancement of Science. 
This address is published in Science, new series, Vol. 30, pages 289-302, September 
3, 1909. 

* We assume that as the substance A cools off, the substance B, only, is heated. 



m 


JOULES TO HEAT ONe 






/ 


1 


QRJ 


>MOF 


/ATER 








/ 




120 












/ 


i 














/ 
















/ 


/ 
















/ 








80 










/ 
























60 






/ 
















/ 
















j 


/ 












10 


















so 


; 


1 














/ 


















/ 


TEMPI 


RATUI 


E RISE 


(hyd 


iOGEN 


SCALEJ 


»" 



10^ 



J5' 20' 85' 
Fig. 89. 



30^ 85*= 



THE FIRST LAW OF THERMODYNAMICS. 



125 



82. The heating of water by the dissipation of mechanical 
energy. — The relation between the amount of work dissipated in 
heating water and the rise of temperature produced has been 
determined with great care by H. A. Rowland. f His results are 
given in the accompanying table and are shown graphically in 
Figs. 89 and 90. The ordinates of the curve in Fig. 89 represent 
the work in joules required to raise the temperature of one gram 

































\, 












En 


Jrgyr 
of or 


equir 

le gra 


3d to 
m of 


raise jthe te 
vsrter one t 


mp eratu re 
egree 




\ 


\ 














at va 


riaus 


temp 


eratu 


•es 






-4.20J 


N, 

oules 


\ 






























\, 






















4.19J 


oules- 






\ 


\ 




























\ 


^ 
















1.18 J 














X 


\, 
































^^ 








^ 
























^ 









aO 15 20 35 aO 

TEMPERATURE CHycJrogen Scale) 

Fig. 90. 



35 



of water from 0° C. to /° C. of the hydrogen scale. This same 
quantity of work is given in the table in the column headed K. 
The curve in Fig. 89 is very nearly a straight line. Therefore, 
jor most practical purposes the amount of energy required to heat 
water may he taken to he proportional to the rise of temperature, 
and the amount of heat required to raise the temperature of unit 
mass of water one degree may be very conveniently used as a 
unit of heat. 

t Rowland's experiments, which were carried out in 1879, are described in the 
Proceedings of the American Academy of Arts and Sciences, new series, Vol. 7. A 
good description of this work of Rowland's and of the work of other experimenters 
in the same field, is given on pages 267-286 of Edser's Heat for Advanced Students. 



126 



THE THEORY OF HEAT. 



TABLE* 

Rowland's Determination of the Work Required to Heat Water. 



Temperature rise from o° C. to 


Energy in joules to heat one 
gram of water. 


(t) 


(£) 


5° 


21.040 


10° 


42.041 


15° 


63.005 


20° 


83.935 


25° 


104.834 


30° 


125.708 


35° 


146.745 



The calorie is the amount of heat required to raise the tem- 
perature of one gram of water one centigrade degree. In accurate 
work the calorie is understood to be amount of heat required 
to raise the temperature of one gram of water from 14^.5 C. to 
15°. 5 C. and it is equivalent to 4.1893 joules. For most practical 
calculations, however, the calorie is taken to be 4.2 joules. 

The British thermal unit is the amount of heat required to 
raise the temperature of one pound of water one Fahrenheit 
degree,! and it is equivalent to about 778 foot-" pounds" of 
energy or work. 

83. Measurement of heat. — An amount of heat, for example, 
the amount required to melt a gram of ice, or to produce a given 
rise of temperature of a gram of lead, is measured when the 
amount of work required to produce the effect has been deter- 
mined. This measurement may be made by the direct deter- 
mination of the work required to produce the given effect. The 
accomplishment of this method of heat measurement is, however, 
very tedious and subject to a very considerable error. J This is 

* From Rowland's results reduced by W. S. Day to the hydrogen scale {Physical 
Review, Vol. VIII, April, 1898). 

t To be precise it should be stated that the British thermal unit is the amount 
of heat required to raise the temperature of one pound of water from, say, 59°.S F. 
to 60°. 5 F. ; but this degree of precision is not demanded by the steam engineer, 
and it is the engineer, chiefly, who uses the British thermal unit. 

X The work spent in any portion of an electric circuit can be measured with con- 
siderably accuracy and it can be easily applied to the accomplishment of any given 



THE FIRST LAW OF THERMODYNAMICS. 127 

partly due to the difficulty of measuring work mechanically and 
partly due to the difficulty of applying mechanical work wholly 
to the heating of a given substance. 

The water calorimeter. — Heat is generally measured by the 
water calorimeter which is a vessel containing a weighed quantity 
of water and arranged to absorb the heat to be measured. For 
example, if the heat produced by the burning of a given quantity 
of coal is to be measured, the coal is burned in a chamber which 
is completely surrounded by the water of the calorimeter and 
the waste gases from the combustion pass through a coil of metal 
pipe also surrounded by the water of the calorimeter. If the 
heat given off by the cooling of a given amount of metal is to be 
measured, the hot metal is dropped into the water of the calorim- 
eter. 

Let M be the mass of the water in grams, let /' be the initial 
temperature of the water, and let t" be the temperature of the 
water after the heat H is absorbed. Then, neglecting the 
heating of the vessel and stirrer and thermometer bulb,* the 
value of H is calculated as follows. 

{a) Accurate calculation. Let E' be the energy in joules 
required to heat one gram of water from 0° C. to t' , and let E" 
be the energy in joules required to heat one gram of water from 
0° C. to t" as given in the table in Art. 82. Then E" — E' is 
the energy in joules required to heat one gram of water from /' 
to t" , and M{E" — E') is the energy in joules required to heat 
M grams of water from t' to /''. Therefore the energy equiva- 
lent of H in joules is: 

H = M{E'' - £0 (49) 

(b) Approximate calculation. The amount of heat required 
to heat water is of course proportional to the mass M of the 

thermal effect, and this electrical method for measuring heat values in energy units 
is perhaps the most accurate method at present available. 

* The method for making allowance for this source of error and for the exchange 
of heat between the calorimeter vessel and the surrounding air may be found in 
any laboratory manual. 



128 



THE THEORY OF HEAT. 



water and it is approximately proportional to the rise of tempera- 
ture as pointed out in Art. 82, so that the amount of heat required 
to heat mass M of water from t' to /'' is : 

H = Mit'^ - t') (50) 

where H is expressed in calories when M is expressed in grams 
and temperatures on the centigrade scale, or in British thermal 
units when M is expressed in pounds and temperatures on the 
Fahrenheit scale. 

84. Specific heat capacity of a substance. — ^The number of 
thermal units required to raise the temperature of unit mass of 
a substance one degree is called the specific heat-capacity or 
simply the specific heat of the substance. Thus the ordinate of 
the curve in Fig. 90 at the 5° point is 4.204 indicating that the 
specific heat of water at 5° C. is 4.204 joules per gram per degree. 
Specific heats are, however, usually expressed in calories per 
gram per degree centigrade or in British thermal units per 
pound per degree Fahrenheit. Thus the specific heat of copper 



































0.4 














y 




"^ 























i 
js 


'V 


/ 


















0.3 










^f 


/ 






) 


' 












< 

UJ 






4 


/ 






















05 


I"" 
I/) 




,^ 


Y 








^^/ 


/ 


















^j 






' 




7 
















0.1 




■i 


V 

Zl 






"Sv;; 


^ 




















7 










































TEM 


=ERA 


PURE 

1 i\mA 


S .. 













200 



J00° 600? 

Fig. 91. 



800' 



is about o 093 calories per gram per degree centigrade or 0.093 
British thermal units per pound per degree Fahrenheit. 

The specific heat of a substance usually varies considerably 



THE FIRST LAW OF THERMODYNAMICS. I29 

with temperature. Thus the ordinates of the curve In Fig. 90 
show the specific heats of water (in joules per gram per degree 
centigrade) at different temperatures, and the ordinates of the 
curves in Fig. 91 show the specific heats of iron and of crystalHne 
carbon at different temperatures In calories per gram per degree 
centigrade. 

Determination of specific heat. Example. — A copper vessel 
weighing V grams contains W grams of water at t° C. Into 
this vessel K grams of hot copper at 7"° C: is dropped, and the 
whole settles to a medium temperature m° C. All quantities 
so far specified are supposed to be known, and it is required to 
find the specific heat c of copper. 

One gram of copper cooling one degree gives off c calories, 
and K grams of copper cooling from T° to m° gives off 
cK{T — m) calories. The heat thus given off by the cooling 
copper Is used to heat the vessel and the contained water. 

To heat one gram of water one degree requires one calorie, and to 
heat W grams of water from f to m° requires W{m — t) calories. 

To heat one gram of the copper vessel one degree requires c 
calories, and to heat V grams of copper from f to m° requires 
cV{m — t) calories. 

But the heat given off by the cooling of the hot copper is the 
heat that raises the temperature of the vessel and water. Therefore 
we have: 

cK{T - m) = W{m - /) + cV{m - t) (i) 

in which c Is the only unknown quantity so that the value of c 
may be calculated. 

85. Heat of reaction. Heat of combustion. — ^When zinc (Zn) 

dissolves in dilute sulphuric acid (H2SO4 + water), zinc sulphate 

(ZnS04) is formed, there is a rise of temperature of the solution, 

and to bring the solution back to the initial temperature of the 

zinc and acid a certain amount of heat must be taken from the 

solution. This heat amounts to 578 calories per gram of Zn 

or to 377 calories per gram of H2SO4, and it is called the heat of 

reaction of Zn and H2SO4. 
10 



130 THE THEORY OF HEAT. 

In some cases a chemical reaction cools the reacting substances 
so that heat has to be given to the residual substance to bring it 
to the initial temperature of the reacting substances. An ex- 
ample of this kind of reaction is the combination of nitrogen and 
oxygen in the electric furnace. 

A reaction like the dissolving of zinc in sulphuric acid is said 
to evolve or develop heat, and it is called an exothermic reaction. 

A reaction like the combining of nitrogen and oxygen in an 
electric furnace is said to absorb heat, and it is called an endo- 
thermic reaction. 

The most familiar example of an exothermic chemical action 
is ordinary combustion, the burning of fuel; and the number of 
thermal units developed by the burning of unit mass of a fuel is 
called the heat of combustion of the fuel.* Thus the heat of 
combustion of soft-wood charcoal is 7,070 calories per gram. 

Exothermic reactions are common and endothermic reactions 
are unusual at low temperatures, whereas the reverse is true at 
very high temperatures. Thus, living in a cool world we can 
burn fuel to keep warm, and if we lived in a very high temperature 
world we could burn nitrogen to keep cool. 

The breaking up of water vapor to form oxygen and hydrogen 
is a chemical action and it is endothermic, because the combina- 
tion of oxygen and hydrogen to form water is exothermic. Now 
the general rule is that endothermic reactions take place at very 
high temperatures, and, as a matter of fact, water vapor does break 
up into oxygen and hydrogen at very high temperatures with ab- 
sorption of heat. If we lived in a very high temperature world 
we could keep cool by un-burning water! Indeed if it were not 
for the extreme slowness of chemical action at low temperaturesf 
we might in our cool world keep warm by un-burning a substance 
like nitrous oxide. 

* Heats of combustion of a great number of substances are given in Landolt and 
Bernstein's Physikalisch-Chemische Tabellen. 

t There are strong theoretical reasons for believing that any substance like 
nitrous oxide (which is formed by endothermic reaction) decomposes very slowly 
at low temperatures. 



THE FIRST LAW OF THERMODYNAMICS. 131 

86. Conduction of heat. Conductivity. — ^There are three quite 
distinct processes by means of which heat is transferred from 
one place to another, namely, (a) The transfer of heat along a 
metal rod one end of which is in a fire, or the transfer of heat 
through the metal shell of a boiler from the fire outside to the 
water inside. This process is called heat conduction; (b) The 
transfer of heat, as from the sun to the earth, by radiation; and 
(c) The carrying of heat from one place to another by a current 
of hot water, or hot air, or steam. This process is called con- 
vection. In many cases transfer of heat takes place by all three 
of these processes simultaneously. 

Flow of heat through a wall. Conductivity. — The quantity of 
heat H which flows through a wall is proportional to the area a 
of the wall, it is proportional to the difference of temperature 
{t — t') between the two faces of the wall, it is proportional to 
the time r, and it is inversely proportional to the thickness d 
of the wall. Therefore we may write 

H^"^^^ (51) 

where ^ is a proportionality factor whose value depends upon 
the substance of which the wall is made; it is called the heat 
conductivity of the wall. 

The heat conductivity of copper at ordinary room temperature 
is about 0.95, heat being expressed in calories, a in square 
centimeters, {t — t') in centigrade degrees, r in seconds, and 
d in centimeters. Using the same units, the heat conductivity 
of iron at ordinary room temperature is about 0.16, of glass 
about 0.0015, of parafhne about 0.0003, and o^ a layer of flannel 
cloth it is about 0.000035. 

A very poor heat conductor is called a heat insulator, and a 
vessel which is surrounded by a very poor heat conductor is 
said to be thermally insulated to a greater or lesser degree. The 
most complete thermal insulation is the high-vacuum space 
between the double glass walls of the Dewar bulb which is 
familiarly known as the thermos bottle. 



CHAPTER VIII. 

THERMAL PROPERTIES OF SOLIDS, LIQUIDS AND GASES. 

87. Melting points and boiling points.* — When heat is imparted 
to a soHd the temperature rises until the soHd begins to melt; the 
temperature then remains constant until all of the substance is 
changed to liquid; the temperature then begins to rise again 
and continues to rise until the liquid boils; the temperature then 
remains constant until the liquid is entirely changed to vapor 
(pressure being unchanged) ; and then the temperature begins to 
rise again as heat is continually imparted to the substance* 
There are thus two periods during which heat is imparted to a 
substance without producing rise of temperature, namely, when 
the substance is melting and when the substance is boiling under 
constant pressure. 

Examples. — Heat is imparted to very cold ice and the tempera- 
ture rises until the ice begins to melt at 0° C, after the ice is all 
melted the temperature rises to 100° C. (if atmospheric pressure, 
is 760 millimeters of mercury), and after the water is all converted 
into steam the temperature of the steam is raised by further 
addition of heat. 

The melting point of a substance is the temperature at which 
the solid and liquid forms of the substance can exist together in 
thermal equilibrium. This temperature varies but slightly with 
pressure. 

The boiUng point of a liquid at a given pressure is the tempera- 
ture at which the liquid and its vapor can exist together in equi- 
librium. This matter is discussed more fully in the next article. 

Every substance, so far as known, has a definite boiling point 

* Extensive tables of melting points and boiling points are given in Landolt 
and Bornstein, Physikalisch-Chemische Tabellen. 

132 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 133 



at any given pressure;* but crystalline substances, only, have 
definite melting points. Thus, ice crystals and water exist side 
by side in equilibrium at a given temperature, whereas amorphous 
substances, such as glass and resin, seem to have no definite 
melting point. These substances grow softer and softer with 
rise of temperature. 

88. Maximum pressure of a vapor at a Given TEMPERA- 
TURE; and minimum temperature of a vapor at a Given 
PRESSURE. — Consider a cylinder filled with carbon dioxide 
gas, as shown in Fig. 92, and suppose it is kept at a constant 



i mmm/M///////mm 



11 



"^ 



carbon dioxide 
ct constant 

temperature t 



ii )i)ii ,u),j )))^/)frrrjiijjjjM 



P I 



1, 



carbon dioxide 
at constant 
pressure p 



ij,ii))>)>jniin>>>i>>rrTTM'. 



Fig. 92. 



Fig. 93- 



GIVEN temperature while the piston PP is being pushed down- 
wards. 

At first the pressure of the gas increases as its volume decreases 
(in accordance with Boyle's law) ; but when the increasing pres- 
sure reaches a certain value, further decrease of volume does not 
cause further increase of pressure, but results in the condensation 
of a portion of the carbon dioxide into liquid form.f There is a 
limit to the pressure that gaseous carbon dioxide can exert, or at 
which gaseous carbon dioxide can exist, at a GIVEN temperature, 
and if an attempt is made to increase the pressure beyond this 
limiting value a portion of the gas condenses to liquid. 

* Below what is called the critical pressure. See Edser's Heat for Advanced 
Students. 

t If the temperature does not exceed a certain value which is called the critical 
temperature of the substance. See Edser's Heat for Advanced Students. 



134 



THE THEORY OF HEAT. 



Or, consider a cylinder filled with carbon dioxide gas, as shown 
in Fig. 93, and kept at a constant GIVEN pressure {by pushing 
the piston downwards if necessary) while the temperature of the 
whole is slowly decreased. At first the temperature of the gas 
decreases as heat is taken from the arrangement; but when the 
decreasing temperature reaches a certain value, further taking 
of heat from the arrangement does not cause further decrease of 
temperature, but results in the condensation of a portion of the 
carbon dioxide into liquid form. There is a limiting temperature 
below which carbon dioxide cannot exist as a gas at a GIVEN 
pressure, and if an attempt is made to cool the gas below this limiting 
temperature a portion of the gas condenses to liquid. 

In discussing the change of a substance from a liquid to a gas 
or from a gas to a liquid it is customary to speak of the gaseous 
form of the substance as vapor. When the vapor is at its maxi- 
mum pressure for a given temperature or at its minimum tem- 
perature for a given pressure, it is said to be a saturated vapor. A 
saturated vapor cannot be cooled without a portion of it being 
condensed if the pressure remains the same. A saturated vapor 
cannot be compressed without a portion of it being condensed 
if the temperature remains the same. 



TABLE. 

Pressures and Temperatures of Saturated Water Vapor. 
(Boiling Points of Water at Various Pressures.) 



Temp. 


Pressure in 


Temp, 


Pressure in 


Temp. 


Pressure in 


Temp. 


Pressure in 


centimeters. 


centimeters. 


centimeters. 


centimeters. 


-IO°C. 


O.2151 


50° c. 


9.1978 


110° C. 


107-537 


170° C. 


596.166 


o° 


0.4569 


60° 


14.8885 


120° 


149.128 


180° 


754.692 


10° 


0.6971 


70° 


23.3308 


130° 


203.028 


190° 


944.270 


20° 


1-7363 


80° 


35-4873 


140° 


271.763 


200° 


1168.896 


30° 


3-1510 


90° 


52.5468 


150° 


358.123 


210° 


1432.480 


40° 


5-4865 


100° 


76.0000 


160° 


465.162 


230° 


2092.640 



What is said here of carbon dioxide is true so far as known of 
every substance. Thus, water vapor at a given temperature 
cannot exert more than a certain maximum pressure, or at a 
given pressure it cannot be cooled below a certain minimum tem- 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 135 



perature without condensation. The accompanying tables give 
the maximum pressures and minimum temperatures of water 
vapor, and of anhydrous ammonia.* 

TABLE. 

Pressures and Temperatures of Saturated Ammonia Vapor (NH3). 
(Boiling Points of Liquid Ammonia at Various Pressures.) 



Temperature. 


Pressure in atmospheres. 


Temperature. 


Pressure in atmospheres. 


-30° c. 

-20° 

-10° 

0° 

10° 


1. 14 atm. 

1.83 

2.82 

4.19 

6.02 


20° C. 

40° 

60° 

80° 
100° 


8.41 atm. 
15.26 
25-63 
40.59 
61.32 



\ 



^ water at '^ 
"^ atmospheric ""^ 
liT pressure Jr 



The boiling point of a liquid at a given pressure is defined in 
Art. 87 as the temperature at which the liquid and its vapor 
stand together in equilibrium at the given pressure, and this is 
the same thing as the minimum temperature of the vapor at the 
given pressure. The temperature at which a liquid at a given 
pressure boils, in the ordinary sense of that term, meaning the 
formation of bubbles of vapor near the bot- 
tom of the containing vessel, is slightly 
variable; it depends to some extent upon 
the rapidity at which heat is given to the 
liquid, and to some extent upon the absence 
of dust particles and the cleanness of the 
containing vessel. The connection be- 
tween boiling temperature and the minimuni 
temperature of vapor at a given pressure may 
be explained as follows : Figure 94 repre- 
sents a bubble 6* of water vapor or steam 
underneath water at atmospheric pressure. 

Therefore the steam itself must be at atmospheric pressure, t and 
if its temperature (the temperature of the water) is less than that 

* For more extensive tables see Landolt & Bernstein's Physikalisch-Chemische 
Tabellen. 

t As a matter of fact the bubble of steam must be slightly above atmospheric 
pressure because of the weight of the overlying water as explained in Art. 53. 




Si - 



1 



Fig. 94. 



136 • THE THEORY OF HEAT. 

for which steam can exert one atmosphere of pressure, the bubble 
of steam will condense into liquid and collapse. The temperature 
of the water must be at least as great as the minimum tempera- 
ture of water vapor at atmospheric pressure in order that the 
bubble of steam may continue to exist, and if the temperature of 
the water is slightly greater than this the bubble of steam will 
continue to grow in size as more steam is formed at its boundaries. 

89. Superheating* and undercooling of liquids. — When water 
which is free from air and dust is heated in a clean glass vessel, 
its temperature is likely to rise 10 degrees or more above its 
boiling point (corresponding to the given pressure) ; and when 
it begins to boil it does so with almost explosive violence and the 
temperature quickly falls to the boiling point. If pure water is 
cooled in a clean glass vessel, its temperature is likely to fall 
considerably below its normal freezing point; and when freezing 
begins a large amount of ice is suddenly formed and the tem- 
perature quickly rises to the normal freezing point. It seems 
that water cannot change to vapor or to ice except there be some 
nucleus at which the change may begin. Most liquids show 
these phenomena of superheating and undercooling. 

Supersaturation of a salt solution. — When a solution of a salt 
stands in equilibrium in contact with undissolved crystals of the 
salt the solu tion is said to be saturated. A slight cooling of the sat- 
urated solution in the presence of the crystals causes additional 
crystals to be formed and the residual solution is weakened, but 
it is still saturated at the lower temperature. 

When a salt solution free from air and dust is set aside to cool 
in a clean glass vessel it generally becomes supersaturated, and 
crystallization of the dissolved salt takes place suddenly when a 
minute crystal of the same salt is dropped into the solution or 

* The term the superheating of a liquid must not be confused with the term the 
superheating of steam. Superheated steam is unsaturated steam, that is, steam of 
which the pressure is less than the maximum pressure at the given temperature, 
or of which the temperature is greater than the minimum temperature for the given 
pressure. Steam may be superheated by passing saturated steam from a steam 
boiler through a coil of pipe in a furnace. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 137 

when the solution is given a vigorous shake. Supersaturation 
is the same thing as undercooUng. 

90. Variation of boiling (or condensing) temperature with 
pressure. — The boiUng (or condensing) temperature of water 
varies greatly with pressure. Thus at normal atmospheric pres- 
sure the boiling point of water is 100° C, at half-of-an-atmosphere 
the boiling point is about 81° C, and at two atmospheres the 
boiling point is about 122° according to the table in Art. 88. 
This dependence of boiling point upon pressure may be empha- 
sized as follows: Given a vessel of water at 75° C. There 
are two ways to bring the water to boiling, namely, raise the 
temperature of the water to its "normal boiling point" for atmos- 
pheric pressure, or lower the "boiling point" to the actual 
temperature of the water by reducing the pressure. 

Another illustration of the dependence of boiling (or con- 
densing) temperature on pressure is afforded by the ammonia 
refrigerating machine the essential features of which are shown 
in Fig. 95. A system of "cooling pipes" in the cold room and a 



pump 



condensing -__ 

water ^ ''^or 



> water 



c 




valve 



NHj liquid ^ NH^ liquid 



condensing pipes 

temperature 2J 

[hot) 



cock 



Fig. 95. 



cooling pipes 
temperature T 
(cold) 



system of "condensing pipes" in the warm outside air contain 
anhydrous ammonia (NH3), and a pump draws ammonia vapor 



138 THE THEORY OF HEAT. 

out of the "cooling pipes" and forces it into the ''condensing 
pipes" thus maintaining a low pressure in the "cooling pipes" 
and a high pressure in the "condensing pipes." Under these 
conditions the liquid ammonia in the "cooling pipes" continues 
to boil, and the ammonia vapor in the "condensing pipes" con- 
tinues to condense to liquid. The condensed liquid flows back 
into the "cooling pipes" and is used over and over again. The 
usefulness of the refrigerating machine depends on the fact that 
the continued boiling or vaporization of the liquid ammonia in 
the cooling pipes keeps the pipes cold while they take in heat from 
their surroundings. Indeed the continued boiling of a liquid 
requires a continued giving of heat to the liquid, and the con- 
tinued condensing of a vapor requires the continued taking 
of heat away from the vapor. This matter is discussed in 
Art. 95. 

91. Variation of freezing points with pressure. — Any substance 
which expands on freezing (as water does when it changes to ice) 
has its freezing temperature lowered by increase of pressure, and 
vice versa. 

Consider the ice on a pond in moderate winter weather when 
the temperature of the surface of the ice is only a few degrees 
below the normal freezing point. Where the edge of a skate 
rests on the ice there is a small region of very high pressure, 
and in this region the melting temperature is reduced by the 
pressure below the actual temperature of the ice. Therefore the 
ice melts and forms a minute layer of liquid water under the skate, 
and the skate slides over this layer of water with very little 
friction. When the skate passes by, the minute quantity of 
water freezes again because it is, of course, very cold.* A skate 
does not slide very easily over extremely cold ice because the 

* Suppose for example the ice is at — 3° C. and that the pressure is increased 
to a value which lowers the freezing point to — 5° C. Then in the high-pressure 
region the ice at — 3° C. is much warmer than its melting point, and it quickly 
melts, and the heat absorbed in melting (see Art. 95) lowers the actual temperature 
of ice and water to — 5° C. That is to say, there is a momentary melting and 
cooling of ice and water under the skate. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 139 

pressure does not lower the freezing point to the actual tempera- 
ture of the ice, and therefore no water forms under the skate 

The cohesion of small particles of ice when pressed together, 
as in the packing of a snow ball, is due to the melting of the ice 
at the points of contact where the pressure is great, and the 
immediate freezing of the resulting water as it flows out of the 
small regions of high pressure. 

92. Boiling points and melting points of solutions. — ^When a 
moderately dilute solution of common salt in water freezes, pure 
ice is formed, and the whole of the salt is left in the residual 
liquid. When a solution of salt in water boils, pure water vapor 
is formed, and the whole of the salt is left in the residual liquid- 
In every such case, namely, when the dissolved substance does 
not pass off with the water vapor or crystallize with the ice, the 
boiling point of the solution is higher than the boiling point of 
pure water and the freezing point of the solution is lower than 
the freezing point of pure water.* 

93. The freezing of solutions and of metallic alloys. — There 
are usually two distinct ways in which a solution can freeze, 
namely, {a) By the deposition of crystals of ice, and (b) By the 
deposition of crystals of the dissolved salt. The following ex- 
ample will make this clear. 

Freezing of sodium chloride solution. — When a weak solution 
of sodium chloride is cooled, it freezes by the formation of 
crystals of pure ice, the residual liquor becomes richer and richer 
in sodium chloride, and the freezing point lowers more and 
more. Thus the ordinates of the curve // in Fig. 96 show the 
freezing points, the ice points, of sodium chloride solutions of 
various degrees of concentration as indicated by the abscissas. 

When a very strong solution of sodium chloride is cooled, it 
freezes by the deposition of crystals of pure sodium chloride, the 

* A good discussion of this subject may be found in Whetham's Theory of Solu- 
tion, Cambridge University Press, 1902. It is also discussed at considerable length 
in Nernst's Theoretical Chemistry (English translation, Macmillan & Co., London, 
1916), and in Jones's Physical Chemistry (The Macmillan Co., New York, 1902). 



140 



THE THEORY OF HEAT. 



residual liquor becomes less and less rich in sodium chloride, and 
the freezing point lowers more and more. Thus the ordinates 
of the curve 55 in Fig. 97 show the freezing points, the salt 
points, of sodium chloride solutions of various degrees of con- 
centration. 

The curve II in Fig. 96 is called the ice curve, and the curve 
55 in Fig. 97 is called the salt curve. Figure 98 shows both 



I 



CIS 

•S 



0° 








-10 








-.20 


\ 


\ 








V 





30 



0° 








—10 




5/ 




-20 




/ 








« 





percentage of salt 

Fig. 96. 



10 20 30 
percentage of salt 

Fig. 97. 



o» 








-10 


/ 


i 




—20 




J 






u- 


Iy 





10 20 30 
percentage of salt 

Fig. 98. 



curves // and SS, and the point of intersection U is called 
the eutectic point for the given solution (sodium chloride in water) . 
There is, of course, a definite temperature corresponding to U 
and a definite percentage strength of solution corresponding to 
U. The former is called the eutectic temperature and the latter 
is called the eutectic mixture. 

The dotted portion of SS corresponds to an unstable under- 
cooled condition in the absence of ice crystals but with salt 
crystals present, and the dotted portion of II corresponds to 
an unstable undercooled condition in the absence of salt crystals 
but with ice crystals present. A solution of sodium chloride 
cannot exist stably at any temperature below the eutectic tem- 
perature, and therefore the eutectic temperature is often called 
the minimum freezing temperature. It is about 22° C. below 
the normal freezing point of water, and the eutectic mixture is a 
solution containing about 23.5 per cent, of salt. When the 
eutectic mixture freezes, ice crystals and salt crystals are de- 
posited simultaneously, the solution remains unchanged in com- 
position as more and more of it freezes, and the freezing tempera- 
ture stands permanently at the eutectic temperature. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 141 

The phenomena of freezing of fused mixtures of salts and the 
phenomena of freezing of metallic alloys are similar to the 



3501 



Pireetifoftedd 

<^ §0 7t> 60 5Q 40 



30 



20 10 



300 



e 

a 

I 
I 

as 



250 



150 



100 



^ 



























-.=;;^ 




•^^ 









B. 












X 


C^^ 


















•Si 


















g' 
9*' 




































1; 









10 20 30 40 50 60 70 80 90 100 
per cent of tin 



Fig. 99- 



phenomena of freezing of salt solutions, and the above described 
behavior of a solution of common salt is perhaps the simplest 
case. Thus Heycock and Neville* have found many cases in 
which the freezing of a metallic alloy causes the deposition of pure 
crystals of one metal or the other, in the same way that the 
freezing of a salt solution causes the deposition of pure crystals 
of ice or pure crystals of salt. The freezing point of the residual 
alloy is steadily lowered by the deposition of pure crystals of 
either metal until a certain point (the eutectic point) is reached 
when the residual alloy (the eutectic alloy) continues to freeze 
without further drop of temperature. 

Another case which is slightly more complicated Is exempli- 
fied by alloys of lead and tin, of which the freezing-point dia- 
gram is shown in Fig. 99. f Along the branch AC of the curve, 

* See Nernst's Theoretical Chemistry (Macmillan & Co.), page 402. 
t Taken from a paper by W. Rosenheim and P. A. Tucker, Philosophical Trans- 
actions of the Royal Society, Series A, Vol. 209, pages 89-122, November 17, 1908. 



142 THE THEORY OF HEAT. 

crystals of lead are deposited containing a variable percentage 
of tin ranging from pure lead at A up to 12 atoms of tin to 
88 atoms of lead as the eutectic point C is approached; along 
the branch BC of the curve, crystals of tin are deposited con- 
taining a variable percentage of lead ranging from pure tin at 
B up to one atom of lead to 500 atoms of tin as the eutectic point 
is approached. The crystals of lead containing a variable per- 
centage of tin and the crystals of tin containing a variable per- 
centage of lead are called solid solutions. 

Figure 100 is a melting-point diagram of alloys of copper and 
magnesium.* These alloys present three eutectic points as indi- 
cated in the figure and four so-called distectic points, or points 
of maximum freezing temperature. The first third of this dia- 
gram, between distectic points i and 2, is a melting-point dia- 
gram of mixtures of pure magnesium and the chemical compound 
Mg2Cu; the middle portion of the diagram, between distectic 
points 2 and 3, is a melting-point diagram of mixtures of the 
two chemical compounds Mg2Cu and MgCu2; and the last 
third of the diagram, between distectic points 3 and 4, is a melt- 
ing-point diagram of mixtures of the chemical compound MgCu2 
and pure copper. 

When a cast metal is slowly cooled, the outside portions of the 
casting differ very considerably in composition from the interior 
portions of the casting; any substance which is present in the 
metal in small quantity tends to collect in the central parts of 
the casting. t 

94. The use of ice and salt as a freezing mixture. — Ice in a strong 
solution of common salt has a very low melting point, 15 or 20 
degrees below zero centigrade (see Fig. 98). Therefore ice mixed 
with salt falls to a temperature of 15 or 20 degrees below zero 
centigrade, and stands at that temperature (if there is no change 
in the concentration of the brine which bathes the ice, that is if 

* From a paper by G. Urazov, abstracted in the Chemische Centralhlatt, page 
1038, for the year 1908. 

t A good introduction to the study of metallic alloys is Ibbotson's translation 
of Goerens' Introduction to Metallography, Longmans, Green & Co., 1908. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 143 



there is an excess of undissolved salt) until all of the ice is melted 
by heat absorbed from surrounding objects. A vessel of pure 
water or cream surrounded by a mixture of ice and salt gives 
off heat to the very cold mixture until the water or cream is 
frozen. The sprinkling of salt on ice or snow in the winter time 

a^m per cent of 3^ 




f 

I 



1<X} 



Ctom9 per cent of -Cu 

Fig. 100. 

does not, as commonly supposed, melt the ice; it lowers the 
melting point below the temperature of the surroundings (if this 
is not more than 15 or 1 6 degrees below zero centigrade) and the 
ice is melted by the heat abstracted from its surroundings. 



144 



THE THEORY OF HEAT. 



Many substances cause a drop of temperature (an absorption 
of heat) when they are dissolved in water even when no ice is 
present. Thus every photographer is famiHar with the very per- 
ceptible cooling which is produced when "hypo" (sodium hypo- 
sulphite) is dissolved in water. 

95. Latent heat of fusion and latent heat of vaporization. — 

When heat is imparted to a substance which is at its melting point 
or at its boiling point, a portion of the substance is melted or 
vaporized and the temperature remains unchanged. The number 
of thermal units required to change unit mass of the solid sub- 
stance at its melting point into liquid at the same temperature 
is called the latent heat of fusion of the substance. The number 
of thermal units required to change unit mass of a liquid* at its 
boiling point into vapor at the same temperature is called the 
latent heat of vaporization of the liquid. 

The boiling point of a substance varies greatly with pressure, 
and the latent heat of vaporization varies greatly with the tem- 
perature of the boiling point of the given substance. Thus, the 
latent heat of vaporization of water is 1043 British thermal units 
per pound at a pressure of one "pound" per square inch (abso- 
lute) and at a temperature of 102° F., whereas it is 965.7 British 
thermal units per pound at standard atmospheric pressure and 
212° F., and it is 844.4 British thermal units per pound at 200 
''pounds" per square inch (absolute) and 381°. 6 F. 

TABLE. 





Melting 
point. 


Latent heat of 

fusion, calories 

per gram. 


Boiling point at 

atmospheric 

pressure. 


Latent heat of 
vaporization, cal- 
ories per gram. 


Water 


0°C. 

327 

-39-5 

115 


80 

5.86 
2.82 
9-36 


100° C. 
78.3 

1500 about 

357 

444-7 

34-9 

46.8 

61. 1 


536 


Alcohol 

Lead 


209 


Mercury 


62 


Sulphur 




Ether 


91 

86.6 

58.5 


Carbon bisulphide 

Chloroform 



* In some cases the substance changes directly from the solid form to vapor 
without passing through the liquid state. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 145 

The accompanying table gives freezing and boiling points and 
latent heats of fusion and vaporization of a number of sub- 
stances at standard atmospheric pressure. 

96. Critical states. — When a liquid and its vapor (confined in a vessel) are 
heated, a portion of the liquid vaporizes, the pressure is increased, the density of 
the vapor increases and the density of the liquid decreases.* When a certain tem- 
perature is reached, the density of the liquid and the density of the vapor become 
equal, and the vapor and liquid are identical in their physical properties. This 
temperature is called the critical temperature of the liquid, and the corresponding 
pressure is called the critical pressure. The heat of vaporization of a liquid is 
less the higher the temperature (and pressure) at which vaporization takes place 
and it becomes zero at the critical temperature.f 

97. Pressure of mixed gases. — When two or more gases are 
mixed in a vessel, the total pressure is equal to the sum of the 
pressures which each component gas would exert if it occupied 
the vessel alone (Dalton's Law). For example, if the amount 
of air in a vessel is such that it alone would exert a pressure p 
and if the amount of water vapor in the vessel is such that it 
alone would exert a pressure w, then the mixture will exert a 
pressure p + w.t 

A result of Dalton's Law is that a definite portion of the total 
pressure of a mixed gas may be considered to be due to each of 
the component gases of which the mixture is made. Thus the 
total pressure of the atmosphere is due in part to the nitrogen, 
in part to the oxygen, in part to the carbon dioxide, in part to 
the water vapor, in part to the argon, etc., of which the atmos- 
phere is a mixture. 

98. Evaporation versus boiling. — It is a common observation 

* This statement may not be exactly correct in some cases. The densities of 
liquid and vapor become more and more nearly equal in every case. 

t A good discussion of the subject of critical temperatures and pressures including 
the celebrated experiments of Andrews on carbon dioxide is given in Edser's Heat 
for Advanced Students, -psiges 201-219. An introduction to van der Waal's theory 
of corresponding states is given in Edser's Heat for Advanced Students, pages 304- 
314. A very full discussion of van der Waal's theory of corresponding states is 
given in Nernst's Theoretical Chemistry, pages 224-230. Macmillan & Co., Lon- 
don, 1904. 

t This statement is very nearly true, the degree of approximation being about 
the same as in the case of Boyle's Law and Gay Lussac's Law. 
II 



146 THE THEORY OF HEAT. 

that water evaporates into the air at temperatures far below 
100° C. A liquid at a given temperature continues to evaporate as 
long as the pressure of its vapor is less than the maximum pres- 
sure its vapor can exert at the given temperature. This is true 
whether the space above the liquid is filled with vapor alone or 
with vapor mixed with any gas at any pressure. For example, 
water vapor can exert a pressure of 355 millimeters of mercury at 
80° C. and if a vessel at 80° C. contains water, the water will 
vaporize until the pressure of the water vapor in the vessel is 355 
millimeters. If the vessel contains nothing but water vapor then, 
of course, the total pressure will be 355 millimeters when equilib- 
rium is reached. If the vessel contains dry air at atmospheric 
pressure, some of the air will be driven out by the vapor which is 
formed, and when equilibrium is reached the water-vapor pressure 
in the vessel will be 355 millimeters and the air pressure will be 405 
millimeters, making a total of 760 millimeters.* If the vessel is 
filled with dry air at any pressure p and suddenly closed before 
any perceptible amount of water vapor is formed, then water 
vapor will form until the total pressure is ^ -f 355 millimeters, 
p being the pressure due to the air alone and 355 millimeters 
being the pressure of the water vapor. 

99. Atmospheric moisture. Hygrometry. — Dew Point. The 
dew point is the temperature to which the atmosphere must be 
cooled in order that the water vapor which is present may be 
saturated. Further cooling of the atmosphere would cause some 
of the moisture to condense. 

Vapor pressure. That part of the pressure of the atmosphere 
which is due to the water vapor which is present is called the 
vapor pressure. This pressure varies from nearly zero to 30 
millimeters, or more. 

Absolute humidity. The amount of water in the air, usually 
expressed in grams of water per cubic meter of air, is called the 
absolute humidity of the air. The absolute humidity varies 
from one gram of water, or less, per cubic meter of air on a very 

* The outside air pressure is assumed to be 760 millimeters. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 147 

cold, dry winter's day to 30 or 35 grams of water per cubic meter 
of air on a moist summer's day. 

Relative humidity. The amount of water in the air expressed 
in hundredths of what the air would contain if it were saturated 
at the given temperature is called the relative humidity. When 
the relative humidity is low, the air is said to be dry; when the 
relative humidity is high, the air is said to be moist, irrespective 
of the actual amount of water which is present. For example, 
20 grams of water per cubic meter would correspond to a relative 
humidity of about 60 per cent, on a warm summer's day and the 
air would seem to be extremely dry, whereas about 5 grams of 
water per cubic meter of air would saturate the air at 0° C. and 
the air would seem extremely moist. 

The method usually employed for the determination of the 
hygrometric elements (dew point, pressure of vapor, absolute 
humidity, and relative humidity) is by use of wet and dry bulb 
thermometers, from the readings of which the various quantities 
may be determined from empirical tables. Such tables are 
published by the United States Weather Bureau.* 

100. Transformation points of solid substances. — Everyone is 
familiar with the three forms in which water can exist, namely, as 
a vapor or gas, as a liquid, and as a solid; but recent researches 
have shown that water can exist as a solid in several distinct 
crystalline forms. Ordinary ice changes to these several forms 
in succession as it is cooled, each change or transformation takes 
place at a definite temperature (at a given pressure), and these 
temperatures are called transformation points or transformation 
temperatures. 

This matter may be illustrated by considering pure iron as 
follows. A crucible filled with melted pure iron is allowed to 
cool, its temperature is observed at stated times, and the ordinates 
of the curve in Fig. 10 1 show the decreasing temperature as a 
function of the time. As the crucible loses heat its temperature 
drops until it begins to freeze, then the temperature remains 

* Weather Bureau Bulletin No. 235. Price, 10 cents. 



148 



THE THEORY OF HEAT. 



constant until all of the iron is frozen. Pure iron shows two 
other constant temperature stages of short duration, one at 
880° C. and one at 780° C. In fact solid iron exists in three 



1600 



1500 



1400 



1300 



1200 



nod 



's 

§ 900 



800 



700 



60Q 



500 



\nuid 


1 
iron 

freesino^oint 1505 ''C 
























































\t' 


ron fsoi 


id) 


















































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886X 


-N 


s/S^in 


nfsolic 


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~78dX 


' ~ "^^^ 


\ 




















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Octroi 


ifaolid^ 


















X 


\ 





10 



20 



30 



40 



50 



60 



70 



8q 



90 



100 



minutes 

Fig. 1 01. 

distinct states or forms which are called 7-iron, /3-iron and a-iron, 
respectively. Let us consider what takes place at 880° C. for 
example, and let us compare it with what takes place when the 
iron freezes at 1505° C. 

The physical properties of a The physical properties of 

substance change when the iron change when the iron 

substance changes from liquid changes from the 7 form to 

to solid. the /3 form. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 149 

The volume of a substance The volume of Iron changes 

changes when the substance when the iron changes from 

changes from a liquid to a solid, the 7 form to the ^ form. 

A certain amount of heat A certain amount of heat 
(latent heat) is given off with- (latent heat) is given off with- 
out any change of temperature out any change of temperature 
when a substance changes from when iron changes from the 7 
liquid to solid. form to the /3 form. 

The change of a substance The change from 7 iron to 

from liquid to solid takes place j5 iron takes place at a definite 

at a definite temperature, pres- temperature, pressure being 

sure being given, and this tem- given, and this temperature is 

perature is called the freezing called a transition temperature 

point of the substance. or point of the iron. 

Impurities in iron (like salt in water) generally lower the 
freezing point of the iron, and also lower the transition tempera- 
tures. Thus the curve in Fig. 102 is the cooling curve of a 
crucible containing melted cast iron which contains five or six 
per cent, of carbon. In this case freezing begins at about 
1150° C. (about 350 degrees lower than pure iron), and the freez- 
ing temperature drops to about 1120° C. as freezing progresses. 
Compare this with the freezing of a salt solution as shown in 
Fig. 96. Below 1100° C. the iron is solid, and but one transition 
temperature seems to exist, namely, at 700° C. Both transition 
temperatures in Fig. 10 1 have been lowered by the carbon, and 
if both exist in Fig. 102 they are so close together as to show 
themselves as one. 

1 01. The recalescence of steel. — ^The phenomenon of under- 
cooling shows itself in an interesting way in the transition of steel 
from the 7 form to the a form (the two transition tempera- 
tures in pure iron seem to be a single transition in steel as in 
cast iron), and is called recalescence. To understand the recales- 
cence of steel let us consider the cooling curve of a clean vessel 
of pure water as shown in Fig. 103. When the water reaches 



150 



THE THEORY OF HEAT. 



the freezing point it does not begin to freeze, but cools below the 
freezing point (undercools) as indicated by the dotted line and 
when freezing does begin the temperature rises suddenly to 



1300 



1200 



11(50 



TOOO 



0^ 

I 

s 

6^ 800 

5 



700 



^00 



500 













a"* 


6 


■ 






■ 














\ 










\ 










^ 


\ , 










\ 


\ 








• 


\ 



10 



30 



40 



50 



minutes 

Fig. 102. 

0° C. When a piece of hot steel is cooled, the change from 
7-form to a-form does not take place when the transition tem- 
perature is reached, but the 7-iron cools below the transition 
temperature, and when the change to a-iron does begin it takes 
place rapidly and the temperature rises suddenly on account of 
the heat which is developed by the change. The recalescence 
of steel can be shown by heating a piano steel wire to a bright 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS, GASES. 151 



^ 

5 

M 


\ 


c 












\, 


^ivater 










\ 










& 




N 


\e 








^ 


cC 




\d 




d 















^ 


•5? 












\ 



axis of time 

Fig. 103. 



red heat by an electric current and then allowing the wire to cool. 
When the cooling wire reaches a low 
red heat the retarded transforma tion 
occurs, and a very decided momen- 
tary increase of brightness is seen 
due to a momentary rise of tem- 
perature. The wire also suddenly 
lengthens when the transformation 
takes place as may be seen by 
stretching the wire horizontally and 
observing the variation in the sag. 

102. Retarded transformations. The hardening of steel. — 

A familiar example of an almost permanently retarded trans- 
formation is afforded by common molasses candy. The crystal- 
lization of syrup is a very slow process because, apparently, of the 
viscosity of the syrup. If the syrup is cooled very, very slowly the 
crystallization takes place at the true freezing temperature, but 
if the syrup is cooled quickly it does not have time to crystallize, 
and the result is the well known molasses candy. Paradoxical 
as it may seem, the syrup when it is cooled suddenly does not 
have time to "freeze" but remains in that physical modification 
which is the stable modification at high temperatures. If, how- 
ever, molasses candy is allowed to stand for some months the 
"freezing" gradually comes about, transforming the substance 
into the crystalline modification which is stable at low tempera- 
tures. 

Retarded transformations occur in the hardening of steel. 
At a high temperature steel settles to thermal equilibrium with 
a certain crystalline structure, that is, with the iron in a certain 
modification and with certain crystalline compounds of Iron and 
carbon present. If the steel Is very slowly cooled the various 
transformations take place at approximately the true transition 
temperatures, and we have what is called annealed steel which is 
the stable form of steel at low temperatures. If, however, the 
steel is cooled very quickly the transformation from one modlfica- 



152 THE THEORY OF HEAT. 

tion to another does not have time to take place, the form or 

modification of the steel which normally exists and is stable at 

high temperatures is left in existence at ordinary temperatures, 

and we have the familiar hard form of steel. Hardened steel is 

an unstable modification and it tends gradually to change to the 

stable modification (soft annealed steel).* This change is greatly 

hastened by a slight rise of temperature. Thus hard steel is 

tempered by heating it slightly for a short time. 

* Extremely hard steel gradually softens. See paper by Carl Barus, Physical 
Review, Vol. XXIX, pages 516-524, December, 1909. 



CHAPTER IX. 

THE SECOND LAW OF THERMODYNAMICS OR THE LAW OF 

ENTROPY. 

103. The second law of thermodynamics. — The principle of the 
conservation of energy in its broadest sense, which covers heat 
effects, is sometimes called the first law of thermodynamics, and 
it is one of the great generalizations of physics. Another entirely 
distinct generalization, one which is most deeply seated in the 
intuitions of all men, is the second law of thermodynamics, or the 
law of entropy, as it is sometimes called. No one can have any 
sort of grasp of the philosophy of physics who does not have 
some degree of clear understanding of both these great generaliza- 
tions. The principle of the conservation of energy is discussed 
in Arts. 47, 48, 80 and 81 ; and the law of entropy is discussed in 
this chapter. 

Necessity of a general point of view. — Consider the successive 
stages of the change which takes place when a system of sub- 
stances is settling to a state of thermal equilibrium, for example, 
consider the successive stages of the change which takes place 
when a piece of red hot iron is dropped into a pail of water, and 
imagine these stages to follow each other in a reverse order. 
If such a thing could be it would mean that the piece of cold 
iron would become red hot and jump out of the pail. No 
one every saw such a thing happen, and the postulate that such 
a thing never can happen is called the second law of thermo- 
dynamics. 

The bald statement that a piece of iron lying quietly in a 
pail of cold water cannot become red hot spontaneously and 
jump out of the pail carries with it no general idea, and to say 
that "nothing of the kind" can take place is too vague to convey 
any meaning at all. Therefore it is evident that a general 

153 



154 THE THEORY OF HEAT. 

statement of the second law of thermodynamics cannot be 
made until we have developed a point of view which will per- 
mit us to speak intelligibly and definitely, but in general 
terms, of all kinds of spontaneous or impetuous action. 

104. Reversible processes. — A substance in thermal equi- 
librium is generally under the influence of external agencies. 
Thus surrounding substances confine a given substance to a 
certain region of space, and they exert on the substance a definite 
pressure; surrounding substances are at the same temperature 
as the given substance; surrounding substances may exert con- 
stant electric or magnetic influences on the given substance; 
and so on. If the temperature of the surrounding substances 
be raised or lowered very slowly, or if the influences they exert 
upon the given substance be made to change very slowly, then 
the given substance will pass through a continuous series of states of 
thermal equilibrium; such a slow change of the substance is called 
a reversible process because the substance will pass through the 
same series of states in a reverse order if the external influences 
are changed slowly in a reversed sense. 

105. Irreversible processes or sweeps. — When a substance is 
settling or tending to settle to thermal equilibrium it may be 
said to undergo a process. Such a process cannot be arrested 
or held at any stage short of complete thermal equilibrium, but 
it always and inevitably proceeds towards that state. Such a 
process may therefore be called a sweeping process or simply a 
sweep. 

A simple sweep is the settling of a closed system* to thermal 
equilibrium. For example, the equilibrium of a batch of gun 
powder in a large room may be disturbed by ignition, and the 
explosion of the powder and the subsequent settling of the 
residual gases to a quiescent state constitute a simple sweep. 
The equilibrium of a gas confined under high pressure in one 

* A closed system is a substance or a set of substances which neither receives 
nor gives off heat, and which neither does work on an outside substance nor has 
work done upon it. 



THE SECOND LAW OF THERMODYNAMICS. 155 

compartment of a two-compartment vessel may be disturbed 
by opening a cock which connects the two chambers, and the 
rush of gas through the opening and the subsequent settHng 
of the gas in both chambers to a state of thermal equilibrium 
constitute a simple sweep. 

Trailing sweep. — When external influences change continu- 
ously and rapidly, a substance Is all the time settling towards 
thermal equilibrium and It never catches up with the changing 
influences, but trails along behind them as it were, and there 
exists what may be called a trailing sweep. Thus the rapid 
heating of water in a tea kettle Is a trailing sweep; the bottom 
of the kettle is always a little hotter than the water. 

Steady sweep. — A substance may be subject to external action 
which, although entirely permanent or unvarying, is incompatible 
with thermal equilibrium, and the substance may settle to a 
permanent or unvarying state which is not a state of thermal 
equilibrium. Such a permanent state of a substance may be 
called a steady sweep. For example, the two faces of a slab or 
the two ends of a rod may be kept permanently at different 
temperatures, and when this Is done the slab or rod settles to a 
permanent or unvarying state. Heat flows through the slab 
or along the rod from the region of high temperature to the region 
of low temperature, and this flow of heat is an irreversible process. 
The ends of a wire may be connected to the terminals of a 
battery or dynamo so that a constant electric current flows 
through the wire, and the heat which is generated in the wire by 
the electric current may be steadily carried away by a stream of 
water or air. Under these conditions the wire settles to an 
unvarying state which is by no means a state of thermal equi- 
librium ; the battery or dynamo does work steadily on the wire 
and this work reappears steadily In the wire as heat. 

1 06. Thermodynamic degeneration. — The most familiar ex- 
ample of a sweeping process is ordinary fire, and, as everyone 
knows, a fire is not dependent upon an external driving cause, but 
when once started it goes forward of itself and with a rush. 



156 THE THEORY OF HEAT. 

It is not exactly correct to speak of a fire as spontaneous because 
this word refers especially to the beginning of a process, whereas, 
we are here concerned with the characteristics of a process 
already begun. Therefore it is better to describe a phenom- 
enon like fire as impetuous because it does go forward of itself. 
Tyndall, in referring to the impetuous character of fire, says 
that it was one of the philosophical difficulties of the eighteenth 
century. A spark is sufficient to start a conflagration, and the 
effect would seem to be out of all proportion greater than the 
cause. Herein lay the philosophical difficulty. This difficulty 
may seem to be the same as that which the biologist faces in 
thinking of the small beginnings of such a tremendous thing 
as the chestnut tree blight in the United States. The chance 
importation of a spore is indeed a small thing, but it is by no 
means an infinitesimal, whereas, under conceivable conditions a 
fire can be started by a cause more minute and more nearly 
insignificant than anything assignable. This possibility of the 
growth of tremendous consequences out of a cause which has 
the mathematical character of an infinitesimal is the remarkable 
thing; and this possibility is not only characteristic of fire but 
it is characteristic of impetuous processes in general. 

Everyone has a sense of the irretrievable aspects of a disaster 
such as the collapse of a bridge, or the wreck of a ship, or the 
destruction of a house by fire; and although such things may be 
forgotten after reconstruction, the fact remains that the destruc- 
tion is absolute, it cannot be undone.* Also everyone has a 
sense of the extreme complexity of detail of a disaster. Imagine 
anyone making the attempt to study the minute details of a 
conflagration, recording the height and breadth and the irregular 
and evanescent distribution of temperature throughout each 
flicker of consuming flame, the story of each crackling sound and 
the extent and character of every sway of timber and wall! 
And consider how utterly useless and uninteresting such a record 

* The material of the house is not destroyed nor is any energy destroyed and 
yet everyone appreciates that something is destroyed. 



THE SECOND LAW OF THERMODYNAMICS. 157 

would be even if it could be made! From a practical point of 
view the important thing in the collapse of a bridge or the 
wreck of a ship or the destruction of a house by fire is the money* 
loss involved, and money loss is the only basis upon which one 
disaster can be definitely compared with another. 

When a charge of gunpowder is exploded in a large empty 
vessel, everything remains after the explosion; all of the ma- 
terial is there and all of the energy is there. And yet the 
powder cannot be exploded a second time! You cannot burn 
your coal twice; and your cake, you cannot eat it and have it! 
The explosion of gunpowder, and the burning of coal, and the 
utilization of food in the body are what we have called sweep- 
ing processes; in detail they are infinitely complicated, and it 
is not only impossible to compare two sweeping processes detail 
by detail but it would be useless even if it could be done. 
However, the impetuous character of a sweeping process sug- 
gests a certain havoc, a certain degeneration, in the substance 
or substances in which the sweep takes place, and if we can 
establish this idea of thermodynamic degeneration as a definitely 
measurable quantity we will have a basis for the comparison of 
any two sweeping processes, namely, in terms of the thermo- 
dynamic degeneration involved in each. Such a quantity does 
in fact exist, and it is usually called increase of entropy. 

To establish the idea of thermodynamic degeneration it is necessary 
to begin with the assumption that every stveeping process does 
involve a definite amount of thermodynamic degeneration {increase 
of entropy); then derive by strict argument the physical consequences 
which necessarily follow from the assumption; and subject these con- 
sequences to the test of experiment. Such a program is, of course, too 
extensive for an elementary text ; but we shall attempt to make 
the meaning of the assumption clear and develop one or two of 
its physical consequences. This development, however, requires 
the use of calculus methods in most cases, and in such cases the 
treatment as given in this text is printed in small type. 

* Other than money values are here ignored for the sake of simplicity. 



158 THE THEORY OF HEAT. 

In a simple sweep the degeneration lies wholly in the rela- 
tion between the initial and final states of the substance which 
undergoes the sweep. This must be true because no outside 
substance is affected in any way by a simple sweep, no work is 
done on or by the substance which undergoes the sweep and 
no heat is given to or taken from it. In a trailing sweep the 
degeneration may lie partly in the relation between the initial 
and final states of the substance which undergoes the sweep 
and partly in the change which takes place in outside sub- 
stances. In a steady sweep, however, it is possible to think of 
the degeneration as lying wholly in the conversion of work into 
heat or in the flow of heat from a high temperature region to a 
low temperature region, or both; because the substance which under- 
goes the sweep remains entirely unaltered."^ Therefore the idea of 
thermodynamic degeneration as a measurable quantity can be 
reached in the simplest possible manner by a careful consideration 
of a steady sweep. In speaking of the havoc which is wrought 
by a sweeping process we have had in mind the impetuous 
character of such a process, and to consider so mild a thing as a 
steady sweep may seem to be a weakening of our argument ; but 
the initial steps of any physical argument should have a vivid 
appeal to sense. It is for this reason that we have directed the 
reader's attention to the quick and spectacular process of burning 
rather than to the slow and invisible process of rotting, although 
it is in the end as bad to lose a house by rotting as it is to lose a 
house by the quick calamity of a conflagration. 

Proposition. — The thermodynamic degeneration which is involved 
in the direct^ conversion of work into heat at a given temperature 

* This matter requires further explanation. Consider, for example, a rod which 
conducts heat steadily from a hot region to a cool region; to eliminate all changes 
of state outside of the rod we must imagine the hot region and the cool region 
each to consist of an infinite amount of substance. As another example consider 
the steady generation of heat in a wire by an electric current; in this case we may 
imagine an ideal, one-hundred-per-cent. -efficient dynamo driven by a very heavy 
fly wheel or by a heavy weight and cord so that purely mechanical changes, only, 
take place outside of the wire. 

t By a sweeping process. 



THE SECOND LAW OF THERMODYNAMICS. 1 59 

is proportional to the quantity of work so converted. This proposi- 
tion may seem to be self-evident, but it is not, because it is 
meaningless until we connect it definitely with something phys- 
ical. Let us consider, therefore, a steady flow of electric current 
through a wire from which the heat is carried away by a stream 
of water so that everything remains unchanged as time elapses. 
Such a process is steady, that is, it remains exactly the same 
during successive intervals of time, and therefore any result of 
the process must be proportional to the elapsed time. Thus the 
amount of thermodynamic degeneration must be proportional 
to the elapsed time and the amount of mechanical energy con- 
verted into heat is proportional to elapsed time. Therefore the 
amount of thermodynamic degeneration, </>, is proportional to 
the amount of work, Wj that is degenerated. That is, we may 

write : 

<f) = 7nW (52) 

where w is a constant wnose value depends only on the tem- 
perature of the body in which the degenerated energy appears 
as heat. 

The higher"^ the temperature the smaller the value of m. This 
is evident from the following considerations: Let a quantity of 
work be degenerated into heat at a certain temperature, and let 
the heat so produced flow to a region of lower temperature. 
This flow is a sweeping process and it must, according to our 
assumption, involve an additional amount of thermodynamic 
degeneration. But the final result could be reached directly 
by the degeneration of the original amount of work into heat 
at the lower temperature, and if our assumption as to the ex- 
istence of a quantity called thermodynamic degeneration is to 
be of value, we must assume that the amount of thermodynamic 
degeneration is the same for any two ways in which the same final 
result is reached from the same initial conditions. Therefore the 
lower the temperature the greater the value of </> in equation 

* The recognition of what we speak of as higher and lower temperatures does 
not depend upon any method for measuring temperature. 



l6o THE THEORY OF HEAT. 

(52) and the smaller the value of m, the amount of work 
degenerated into heat being given. 

As yet we have not agreed upon any fundamental measure of 
temperature.* Therefore we are at liberty to adopt the quantity 
i/m as our measure of temperature inasmuch as this quantity 
always has a definite value for any given temperature and inas- 
much as its value is larger and larger the higher the temperature. 
Consequently we may write i/T for w in equation (52), and 

we have: 

W 

= jT (53) 

or, since W is equal to the heat H which has been produced, 
we may write: 

</> = y (54) 

Note. — It is a remarkable and significant fact that the sim- 
plest line of argument concerning thermodynamic degenera- 
tion should make it proportional to elapsed time. Indeed the 
two quantities, time and thermodynamic degeneration, do refer 
to the same condition in nature, to the universal forward move- 
ment of things which we all have come to think of as inevitable, 
and never to be reversed. 

107. Kelvin temperature values,t again. — The definition of 
temperature values which is involved in the above discussion of 
thermodynamic degeneration and which is stated in explicit terms 
in Art. no, is due to William Thomson (Lord Kelvin), and there- 
fore these temperature values are called Kelvin temperatures. In 
Art. 116 it is shown that temperature values as measured by the 
air thermometer are very nearly identical to temperature values 
according to the Kelvin definition. Therefore the temperature T 
as used throughout this chapter is Kelvin temperature as meas- 

* The measurement of temperature as explained in Art. 74 is entirely arbitrary 
and it depends upon the particular gas which is used in the "air" thermometer. 
A fundamental measure of temperature must be independent of the peculiar 
properties of any particular substance. 

t Kelvin temperatures are frequently called "absolute" temperatures. 



THE SECOND LAW OF THERMODYNAMICS. l6l 

ured by the air thermometer. To get Kelvin temperature add 
273° to ordinary centigrade temperature. Thus 40° C. = 313° K. 

108. Statements of the second law of thermodynamics. — 

(a) The thermodynamic degeneration which accompanies a 
sweeping process cannot be directly repaired, nor can it be 
repaired by any means without compensation. 

This is an entirely general statement of the second law and 
the two terms direct repair and compensation must be explained. 

The direct repair of the havoc wrought by a sweeping process 
means the undoing of the havoc by allowing the sweep to perform 
itself backwards, an idea as absurd as the idea of allowing a 
burned house to unburn itself! 

One can, of course, rebuild a burned house, but a certain 
amount of expense is involved. Also the havoc wrought by a 
sweeping process can be repaired, but when such repair is 
finished some other substance is always left in what may be 
called a degenerated state. The original sweep represents a 
certain degeneration of the substance involved, and when the 
effects of the sweep are repaired, even by ideal reversible proc- 
esses, this degeneration is handed along to some other substance. 
This equivalent degeneration of the other substance is what is 
referred to by the use of the word compensation. 

(&) Heat cannot pass directly from a cold body to a hot body, 
nor can heat be transferred from a cold body to a hot body by 
any means without compensation. 

This reference to the direct passage of heat from a cold body 
to a hot body should recall what has been said about a piece of 
iron lying quietly in a pail of cold water and becoming suddenly 
red hot and jumping out of the pail! 

(c) Heat cannot be converted directly into work, nor can heat 
be converted into work by any means without compensation. 

The direct conversion of heat into work means the simple 
reverse of any ordinary sweeping process which involves the 
degeneration of work into heat. For example, work is degen- 
erated into heat in the bearing of a rotating shaft, and every 



1 62 THE THEORY OF HEAT. 

one knows that to reverse the motion of the shaft does not 
cause the bearing to grow cold and the heat so lost to appear 
as work helping to drive the shaft! That would be a rotary 
engine indeed ! 

{d) A gas cannot pass directly from a region of low pressure 
to a region of high pressure, nor can a gas be transferred from a 
low pressure region to a high pressure region by any means 
without compensation. 

Imagine a gas squirting Itself backwards through a nozzle 
into a high pressure reservoir! But the second law of thermo- 
dynamics is the statement of a fact which everyone knows 
coupled with a generalizing clause or postulate which no one 
can understand until some of its consequences are derived and 
tested by experiment. 

Here is one more statement of the second law of thermody- 
namics, the oldest English version of it: 

Humpty Dumpty sat on a wall. 

Humpty Dumpty had a great fall. 

All the King's horses and all the King's men 

Cannot put Humpty Dumpty together again! 

This is perhaps the most sensible of all the statements of the 
second law, for which we will allow it to pass for the moment, 
because it ignores direct repair and refers at once to the most 
powerful of external means. It is important to understand, 
however, that in Humpty Dumpty's case we are concerned 
with structural degeneration, not with the vastly simpler kind 
of degeneration which, for example, takes place when you pro- 
duce turbulence in a pail of water by stirring. Of course the 
water can be easily brought back to its initial condition, but 
a certain compensation is always involved. 

Of all the generalizations of physics the second law of thermo- 
dynamics is the most deeply seated In human experience and 
intuition, and one of the most humorous of children's verses 
refers to the man whose wondrous wisdom enabled him to cir- 
cumvent it by direct repair! 



THE SECOND LAW OF THERMODYNAMICS. 163 

There was a man in our town, 

And he was wondrous wise; 
He jumped into a bramble bush 

And scratched out both his eyes. 
And when he found his eyes were out, 

With all his might and main, 
He jumped into another bush 

And scratched them in again. 

But let us return to the fourth statement {d). A gas can be 
transferred from a low pressure region to a high pressure region 
by means of a pump, and the work that is used to drive the 
pump, even supposing the pump to be frictionless, is all con- 
verted into heat. This conversion of work into heat is the 
compensation for the transfer of the gas as specified. 

Or consider the second statement ih). In an artificial ice 
factory heat is transferred from the freezing room to the warm 
outside air, but the work required to drive the ammonia pump, 
even supposing it to be frictionless, is converted into heat. 

Or consider the third statement (c). An ordinary steam 
engine converts heat into work, but even with an ideal or per- 
fect engine a large amount of heat must be supplied to the 
engine at boiler temperature and a large portion of this heat 
must be let down to the temperature of the exhaust and pass 
out with the exhaust to compensate for the conversion of the 
remainder into work. 

109. The steam engine. — One of the most important conse- 
quences of the second law of thermodynamics is that a steam 
engine working between given boiler temperature and given con- 
denser temperature has a limiting efficiency beyond which it is 
impossible to go whatever the design of the engine and however 
carefully it may be made. Furthermore it can be shown that 
any sweeping process which takes place in the operation of a 
steam engine must necessarily reduce Its efficiency. These two 
consequences of the second law of thermodynamics have served 
as important guides to the steam engine designer as explained 
in Art. iii. 



1 64 



THE THEORY OF HEAT. 



The ideal or perfect engine and its efficiency. — The important 
theorem concerning the maximum possible efficiency of a steam 
engine is estabHshed by an argument based on the assumption 
that the operation of the engine involves no sweeping processes 
of any kind. The ordinary engine does involve such processes, 
but if the engine were frictionless, if it were run slowly, if the 
cylinder were prevented from cooling the steam, and if the steam 
were expanded in the cylinder sufficiently to prevent puffing, 
then no sweeping processes would be involved. Such an ideal 
engine we will call a perfect engine. 

The essential organs of the steam engine are shown in Fig. 104. 
The feed-water pump is arranged to take not only water but 
also a certain amount of uncondensed steam, and the com- 
pression of this mixture of steam and water by the pump con- 



steam 



-^=^ 



Hillll'M'lllil 



engine 
cylinder 



*^^"'^ JM condensing 



\ water 




fire 



water 



water 



Fig. 104. 



denses the steam and raises the feed-water to boiler temperature. 

With the feed-water pump acting in this way no outstanding change 
of any substance is involved in the action of the engine; and, if we 
assume everything to be frictionless, etc., as specified above in 



THE SECOND LAW OF THERMODYNAMICS. 165 

the definition of a perfect engine, then no irreversible or sweeping 
processes are involved in the operation of the engine, and the 
only outstanding effect after the engine has been in operation 
for a while is that a certain amount of heat Hi has been taken 
from the boiler at temperature 7"i, a portion of this heat has 
been converted into mechanical work W, and the remainder of 
the heat H^ has been delivered to the condenser with the 
exhaust steam at temperature T2. 

According to the first law of thermodynamics the work W 
must be equal to Hi — H2, heat being expressed in energy units, 
Therefore we have 

W = H,-m (55) 

For the sake of simplicity of argument the net result of the 
operation of the engine may be thought of as (a) The conversion 
into work of the whole of the heat Hi which is taken from the 
boiler at temperature Ti, and {h) The reconversion of a portion 
H2 of this work back into heat at condenser temperature T^. 

The regeneration^ associated with process (a) is equal to Hi/Ti 
and the degeneration associated with process (6) is equal to H2/T2. 
But since we have assumed that the engine operates without any 
sweeping processes, there can be no degeneration; that is the 
regeneration involved in process (a) must be equal to the de- 
generation involved in process (6). That is, we must have 

Hi H2 . 

Substitute the value of H2 from equation (55) and solve for W, 
and we get 

* This argument may be given in a more concrete and convincing form by using 
the idea of entropy as developed in Art. 112. For the present we must be content 
with this highly abstract form. To convert an amount of work W into heat at 
temperature T involves an amount of degeneration, and to convert the heat back 
into work would, if it were possible, involve the same amount of regeneration. 



I66 THE THEORY OF HEAT. 



(^^)- 



That is to say, the fractional part I — li?i of the heat 

Hi is converted into work by the engine, and the fraction 

■ j; IS called the efficiency of the engine. 

Efficiency of the imperfect engine. — If the operation of a 
steam engine involves sweeping processes of any kind, then the 
thermodynamic degeneration which is associated with process 
(b) above mentioned must be greater than the regeneration which 
is associated with process (a). That is, we must have 

Hi H2 
Ti ^ T2 

or, substituting the value of H2 from equation (55) and solving 
for W, we get 

W < ^^^ -H, 

J^ 1 

Therefore, comparing this inequality with equation (57), we 
conclude that any engine whose operation involves sweeping proc- 
esses of any kind must he less efficient than our ideal or perfect 
engine, boiler and condenser temperatures being given. 

no. The Kelvin definition of the ratio of two temperatures. — 

The definition of temperature value which is involved in equa- 
tions (52) and (53) becomes explicit in equation (56) inasmuch 
as this equation may be written thus : 

Ti^Hi 
Ti Hi 

which is to say, the ratio of two temperatures is the ratio of the 
respective quantities of heat taken in and given out by a perfect 
engine working between those temperatures, as above explained. 
This definition is due to Lord Kelvin. 

III. Conditions which affect the efficiency of a steam engine 
in practice. — A fraction of the heat which is delivered to an 



THE SECOND LAW OF THERMODYNAMICS. 167 

engine is converted into work. In order that this fraction may 
be large, the fraction T1JT2 must be large and sweeping processes 
must be obviated as much as possible in the operation of the 
engine as explained in Art. 109, Ti being the temperature of 
the steam or gas as it begins pushing on the engine piston and 
7^2 being the temperature of the exhaust steam or gas. 

Boiler temperatures in ordinary steam engine practice range 
from about 170° C. to about 200° C. (443° K. to 473° K.), and 
the lowest feasible exhaust steam temperature is about 40° C. 
(313° K.). In the gas engine the initial temperature is the tem- 
perature of the mixture of gas and air immediately after the 
explosion, and may be 1500° C. (1773° K.) or higher; and the 
temperature of the exploded mixture is reduced by expansion to 
six or eight hundred degrees centigrade (873° K. or 1073° K.). 

The sweeping processes which occur in a steam-engine plant 
are as follows: 

(a) Wire drawing. — If the pipes and passages traversed by 
the steam from the boiler to the engine are small, the pressure in 
the cylinder with open ports will be lower than boiler pressure, 
so that the entering steam passes from a region of high pressure 
into a region of low pressure. Also as the cut-off valve closes, 
steam will rush into the cylinder through a narrowing aperture. 
This effect is called wire drawing, and to provide against loss of 
efficiency from this cause, the pipes must be of ample size and 
the cut-off valve must operate very quickly. 

(b) Radiation. — The cooling of pipes and cylinder by the giving 
of heat to surrounding cooler bodies is a sweeping process, and is 
to be obviated as much as possible by covering pipes and cylinder 
with a thick coating of porous insulating material. 

(c) Cylinder condensation. — As a charge of steam In the cylin- 
der expands it cools and cools the cylinder and piston, so that 
when steam is next admitted it heats the cylinder and piston up 
again. This effect cannot be eliminated, but it can be largely 
reduced by providing separate passages for the ingress and egress 
of steam and by using a series of cylinders of increasing size, 



1 68 THE THEORY OF HEAT. 

the smallest cylinder being arranged to take steam directly from 
the boiler and exhaust into the next larger cylinder which in 
turn exhausts into a still larger cylinder, and so on. In this way 
the range of temperature in each cylinder is small and the effects 
of cylinder condensation are greatly reduced. A steam engine 
in which expansion of the steam takes place in two stages (in 
two cylinders) is called a compound engine. A steam engine 
in which the expansion of the steam takes place in three stages 
(in three cylinders) is called a triple expansion engine. 

The loss of efficiency due to cylinder condensation is greatly 
reduced by the use of superheated steam because the exchange 
of heat between the steam and the cylinder walls is very greatly 
reduced when the steam does not condense. Thus S. LeRoy 
Brown has found that heat is imparted to a cool metal surface 
about forty times as fast by condensing steam as by a gas at the 
same temperature. 

(d) Effect of high piston velocity. — If the piston speed is too 
great, the pressure of the expanding steam becomes ineffective 
because the portions of the steam near the moving piston are 
expanded and cooled before the more remote parts of the steam 
are affected. This effect is negligible at the highest piston ve- 
locities which are mechanically feasible. 

(e) Puffing. — The steam at the end of a stroke is usually at a 
pressure which exceeds the pressure in the condenser and it 
rushes through the exhaust port as a sharp puff. This effect can 
be avoided by sufficiently reducing the steam pressure by expan- 
sion, but expansion should never be carried so far as to give a 
force on the piston (due to the steam) less than the frictional 
drag on piston and cross-head. 

(f ) Furnace to boiler. — The most pronounced sweeping process 
which intervenes between the completed combustion and the 
final exhaust of the steam from the engine is the flow of heat 
from the very high temperature of the fire in the furnace to the 
moderately low temperature of the water in the boiler, and the 
greatest waste in the operation of the steam engine in the sense of 



THE SECOND LAW OF THERMODYNAMICS. 



169 



loss of availability of heat for conversion into work is involved 
in this sweeping process. This waste can hardly be avoided in 
the steam engine because of the danger involved in the genera- 
tion of steam at very high pressures (and temperatures) in a 
large boiler. 

Ordinary waste. — ^The greatest items of waste in the ordinary 
sense of actual loss of heat are (a) the incomplete combustion 
of the fuel, and {h) the carrying away of great quantities of heat 
in the flue gases. The economic use of fuel for the production 
of mechanical power requires therefore a properly designed 
furnace and intelligent and careful stoking to insure complete 
combustion, and it requires a sufficient exposure of boiler surface 
and frequent cleaning of the same to facilitate the flow of heat 
from the hot gases into the boiler. 

112. Entropy of a substance in a given state. — ^The idea of 
thermodynamic degeneration as developed in the foregoing dis- 
cussion is highly abstract, it would be much simpler if there 
were a measurable property of a substance such that the difference 
in the measured values of this property before and after a simple 
sweeping process would be the thermodynamic degeneration 
involved in the sweep. Such a measurable property does in 
fact exist and it is called the entropy of the substance. 

The amount of thermodynamic degeneration involved in a 
given change of state of a substance, a change from state or 
condition A to state B, must be assumed to be independent 
of the intermediate stages through which the substance passes 
in going from state A to state B, otherwise the idea of thermo- 
dynamic degeneration amounts to nothing. Therefore, adopting 
the assumption, we may choose arbitrarily a standard state of the 
substance for which a certain quantity </> is taken to be zero, 
and define the value of <j> for any given state of the substance as 
the thermodynamic degeneration involved in changing the sub- 
stance from the standard state to the given state. The quantity 
so defined is called the entropy of the substance in the given state. 

113. Change of temperature of a gas during free expansion. 



1 70 



THE THEORY OF HEAT. 



The liquid air machine. — The expansion of a gas in a cylinder 
against a slowly receding piston is called constrained expansion. 
During constrained expansion a gas does work on the receding 
piston, and the temperature of the gas falls greatly. This cooling 
of a gas during constrained expansion is exemplified by the cooling 
of steam as it expands in a steam-engine cylinder, and by the 
cooling of the air which drives an ordinary pneumatic tool such 
as a rock drill. Not only is a gas cooled during constrained 
expansion, but, conversely, a gas becomes hot when it is com- 
pressed. 

If a piston in a cylinder could move so as to cause an instan- 
taneous increase of volume, the gas in the cylinder would follow 
the piston more or less sluggishly without pushing upon it and 
the expanding gas would do no work. Such expansion would be 
called free expansion. 

Figure 105 represents a vessel having two closed chambers A 

and B. Suppose that gas is con- 
fined in chamber A at tempera- 
ture 7", and suppose the cock C 
to be opened so that the gas can 
expand freely and fill both cham- 
bers A and B. In this case the 
expanding gas does no external work, 
and, after the expanded gas has settled 
to thermal equilibrium {heat being neither given to nor taken away 
from AB), the temperature of the gas is approximately at the same 
temperature as before. 

The question of the rise or fall of temperature of a gas during 
free expansion has been studied with great care by using an 
arrangement in which the gas expands through a small orifice 
or through a porous plug,* and it has been found that a slight 

* The "porous-plug" experiment of Joule and Thomson is described in Edser's 
Heat for Advanced Students, pages 379-372. A very complete discussion of the 
theory of the experiment is given in Buckingham's Thermodynamics, pages 127-137, 
The Macmillan Co., 1900. This experiment has an important bearing on the 
interpretation of temperature values as measured by the air thermometer as may 
be seen from Art. 116. 




Fig. 105. 



THE SECOND LAW OF THERMODYNAMICS. 



171 



change of temperature does usually take place. Hydrogen is 
slightly cooled and oxygen and nitrogen are slightly warmed by 
free expansion. All gases, however, are very considerably cooled 
by free expansion if the initial pressure is very great and the initial 
temperature low. 

Lindens liquid air machine. — ^The cooling effect of free expan- 
sion at low temperatures and at very high pressures is utilized 
in Linde's liquid air machine. This machine operates as follows: 
Air under great pressure (150 to 200 atmospheres) is forced 
through a large coil of small copper tube at the end of which 
it escapes through a fine orifice into a low-pressure region, 
whence it flows back over the coil of copper tube. The expan- 
sion of the air at the fine orifice cools the air slightly; this 
slightly cooled air in flowing back over the coil of copper pipe 
cools the inflowing air, which in its turn is further cooled when 
it passes through the orifice; the inflowing air is then still fur- 
ther cooled in the coil of pipe, and so on, until the temperature 
at the orifice is so greatly reduced that a portion of the air con- 
denses into a liquid which collects in the low-pressure chamber 
and is drawn off at will. 



114. The entropy of an ideal gas. — To derive an expression for the entropy of a 
gas let us consider an ideal gas, a gas which is assumed to conform exactly to 
Boyle's law {pv = a constant) and which is not changed in temperature by free 
expansion. At the start let us have volume v of the gas at pressure p and at 
a given temperature T, and let this gas be carried through two processes (a) and 
(b) which bring the gas back to its initial condition. 

(a) Let the gas expand freely by suddenly increasing its volume by the amount 
Av. The entropy of the gas must increase during this free expansion because free 
expansion is a sweeping process. 

(&) Let the gas be slowly compressed to its initial volume, and let the necessary 
amount of heat AH be taken from the gas to keep its temperature T constant 
during the compression. 

During this process (&) the decrease of volume of the gas is Av, and the work 
AW done on the gas is equal to p-Av.* Furthermore, when the gas has been 
brought back to its initial condition its total energy is, of course, the same as at 
first, so that the total work AW done on the gas must be equal to the total heat 
AH which has been taken from the gas. The net result of processes (a) and (6) 

* This can be shown by an argument similar to the argument used in Art. 65 
for establishing equation (31). 



172 THE THEORY OF HEAT. 

together is the conversion of the amount of work AW into heat (AH) at temperature 
T; therefore processes (a) and (&) together involve an amount of thermodynamic 
degeneration A0 such that 

AW _ p-Av AH 

I rr\ rr% rr% ^ ' 

according to equations (53) and (54). 

But process (6) is reversible and it involves no thermodynamic degeneration 
Therefore the thermodynamic degeneration A</) resulting from both processes is 
due to process (a) alone; and, according to Art. 112, this thermodynamic degenera- 
tion must be equal to the increase of entropy of the gas during process (a) or to the 
equal decrease of entropy of the gas during process ib) — for, inasmuch as 
process (h) brings the gas back to its initial condition it must bring the entropy of 
the gas back to its initial value. Therefore 

p'Av 
Decrease of entropy of the gas during process {h) = (n) 

But Av is understood to be negative, whereas if it were positive [that is, if process 
{b) were constrained expansion] then process (6) would represent an increase of 
entropy of the gas. Therefore 

Increase of entropy of an ideal gas during } ^ , p-Av ..... 

r = A<j) = ■ (ni) 

slow expansion at constant temperature J T 

According to Boyle's law, namely, pv = C, we have p = C/v, and by sub- 
stituting this value of p in equation (iii) we get 

A^=^-^° (iv) 

T V 

But the temperature T is understood to be constant and of course C is a 
constant. Therefore equation (iv) is easily integrated, giving 

C 

<f> = ■- log z; + an unknown function of T (v) 

It is an easy matter to determine the unknown function of T if the specific 
heat of the gas at constant volume is known, but it is not worth while to follow the 
matter here. The important use of the above discussion is that it leads to a 
general expression for the change of entropy of any substance during a reversible 
process. 

115. General expression for change of entropy of any substance during a reversi- 
ble process. — The process (a) as described in the previous article increases the entropy 
of the gas but this process (a) does not involve any change whatever in any outside 
substance. — On the other hand process (b) brings the gas back to its initial condition 
and therefore to its initial entropy, that is, it produces a decrease of entropy of the 
gas amounting to AH IT, according to equation (ii), but process b is an irreversible 
process and it must on the whole involve no thermodynamic degeneration. There- 
fore process b must cause an increase of entropy of the substance which surrounds 



THE SECOND LAW OF THERMODYNAMICS. 



173 



the gas and to which the heat AH is given by the gas during process b. But 
this surrounding substance may be any substance whatever in thermal equihbrium 
at temperature T receiving heat slowly from the gas during process b. 
Therefore the entropy of any substance whatever (in thermal equilibrium at 
temperature T) receiving an amount of heat AH slowly has its entropy in- 
creased by the amount 



A(b = 



(58) 



Change of entropy, only, is physically real, and if one is to assign a definite 
value to the entropy of a given substance in a given state it is necessary to choose 
arbitrarily a standard condition or state of the substance in which state the entropy 
has the value zero, and to define the entropy of the substance in any given state 
by the equation 

given condition 



(59) 



standard condition 



that is, one must devise a reversible process carrying the substance from the 
standard condition or state to the given condition, one must divide this process 
into many small steps, one must determine the amount of heat AH given to the 
substance and the temperature T of the substance during each step, one must 
find the value of the quotient AH/T for each step, and one must find the sum of 
such quotients for all the steps. This sum is the entropy of the substance in the 
given state. 

116. Proposition. — Imagine a gas which conforms exactly to Boyle's law and 
which does not change in temperature during free expansion. Such a gas would 
he called a.n ideal gas. Let t be temperature as measured by an " air " thermometer 
filled with our ideal gas. Then t so measured would be identical to T as defined 
in Arts. 107 and no. 

Proof. — Consider volume v of an ideal gas at pressure p and temperature t. 
Then, since the gas conforms to Boyle's 



law and since t is measured by an 
"air" thermometer filled with the gas, 
we have, according to Art. 75, 



axis of pressure 



pv = Rt 



(i) 



Pz\ ° axis of volume 



Let the gas be carried through four 
processes which bring the gas back to 
its initial condition; these four proc- 
esses being represented by the lines 
ab, be, cd and da in Fig. 106. 

Process i. Beginning at state a 
(pressure p2, volume V2 and tem- 
perature t as represented in Fig. 106), let the gas be compressed to state b at 
constant temperature t. The amount of work, W, done on the gas and the amount 




174 THE THEORY OF HEAT. 

of heat, H, taken from the gas during this process are both given by the equation: 

W = Rt-log- = H (ii) 

This equation is obtained by integrating dW = p-dv from pi to p2, using 
equation (i). 

Process 2. The gas is then heated at constant volume from temperature t to 
temperature t + A^, that is from state h to state c, and an amount of heat A/t 
is given to the gas. 

Process 3. The gas is then expanded from state c to state d at constant 
temperature i + Ai ,• and the amount of work, PF + APF, done by the gas and 
the amount of heat, H + A/?, given to the gas are both given by the equation: 

W +^W = R{t -\-M)'\og~ = H + ^H (iii) 

Vl 

Process 4. The gas is then cooled at constant volume from temperature f + Af 
to temperature t, that is from state d to the initial state a, and an amount of 
heat A/i' is taken from the gas. 

The quantities of heat Ah and A/j' are equal because by free expansion the 
gas might be changed directly from state c to state d, and by free expansion 
the gas might be changed directly from state h to state a, so that the same amount 
of heat must be required to heat the gas from 6 to c as to heat the gas from a 
to d. Therefore let the common value of A/j and Ah' be represented by Ah. 

The increments of the entropy of the gas for each of the four stages are as 
follows, according to equation (58), 



Process i. 



Rt-log"^ 

Vl 

T 



Ah 
Process 2. — , infinitesimals of second order being dropped. 



Process 3. 



m +A^)-iog- 

Vl 

T -\-AT 



Ah 
Process 4. , infinitesimals of second order being dropped. 

In these expressions T and T -{- AT are the thermodynamic or true Kelvin 
temperature values corresponding to t and t + A^ respectively. 

But the total change of entropy of the gas due to the entire cycle is zero because 
the gas comes back to its initial condition. Therefore we have: 

Rt-log- R(t +AO-log- ,. . 

Vl Vl (iv; 

T T -\- AT ~ 

which is easily reduced to the differential equation: 

AT At , V 

-7=- = — (v) 

T t 



THE SECOND LAW OF THERMODYNAMICS. 175 

which, by integration, gives: 

T =kt (vi) 

where k is any constant. Let the value unity be chosen for k, then we have: 

T = t (vii) 

That is, the ideal gas thermometer measures thermodynamic temperature values. 
117. Variation of freezing point with pressure. — One of the most interesting 
physical consequences of the second law of thermodynamics is the relation 

T 

AT = - j{i - w)-Ap (60) 

where Ar is the change (the negative sign means a lowering) of the freezing" point 
of water due to an increase of pressure Ap. The meaning of the other quantities 
are as follows: 

T = 273° is the Kelvin temperature of freezing water at normal atmospheric 
pressure (760 millimeters or 1,013,000 dynes per square centimeter). 

Z, = 80 calories per gram or 3,360,000,000 ergs per gram is the latent heat of 
fusion of ice. 

i = 1.09 cubic centimeters is the volume of one gram of ice. 

w = 1. 000127 cubic centimeters is the volume of one gram of ice water. 

In the following discussion we will speak as if increase of pressure caused a rise 
of freezing point and as if ice were more dense than water {w greater than i) ; 
as a result the necessary algebraic signs will appear in the final equation. 

Consider one gram of water at its freezing point T and at pressure p, the volume 
of the gram of water being w cubic centimeters. The coordinates of the point a 
in Fig. 107 represent p and w so that the point a represents the initial condi- 
tion of our one gram of water. 

Let the one gram of water be carried slowly through four slow changes, i, 2, 3 
and 4 as described below, which bring it back to its initial condition. 

1. Freeze nearly all of the water at temperature T and pressure p by taking 
the necessary amount of heat H2 from it. This freezing reduces* the volume from 
w to i, and this first process is represented by the line ab in Fig. 107. 

2. Increase the pressure to ;^ + Ap without giving heat to or taking heat from 
the substance (ice or water as the case may be). This increase of pressure raisesf 
the freezing point to T + AT. Thus the ice and the minute quantity of residual 
water will be left colder than the freezing point, and as a result the minute quantity 
of residual water will freeze and the latent heat given out by this freezing will warm 
the whole substance up to T -{- AT. This process is represented by the infini- 
tesimal line be in Fig. 107. 

* If this should be the wrong guess the actual measurement of w and i will 
set the matter right. 

t If this should be the wrong guess a negative sign will appear and reverse the 
guess, as it were. 



176 



THE THEORY OF HEAT. 



3. Melt nearly all of the ice at temperature T + AT and pressure p + Ap 
by giving the necessary amount of heat -Hi to it. This melting is accompanied by 
an increase (see above) of volume from i to w (see guess under i), and this process 
is represented by the line cd in Fig. 107. 



axis of pressure 



B. 



g melting at T+^T ^ 

\ i^////{m'lTJJTrh: 

I h freezing at T , <* 






H. 



-^ H 



I 
I 

IP 



axis of volume 



Fig. 107. 



4. Decrease the pressure from p -^ Ap to p without giving heat to or taking 
heat from the substance. This decrease of pressure lowers the freezing or melting 
point from T + AT to T (see guess under 2). Thus the water and the minute 
quantity of residual ice will be left warmer than the melting point, and as a result 
the minute quantity of residual ice will melt and the heat absorbed by this melting 
will cool the whole substance down to T. This process is represented by the 
infinitesimal line da in Fig. 107, and it brings the substance back to its initial condi- 
tion. 

These four processes involve the taking in of heat Hi at the higher temperature, 
the giving out of heat Hi at the lower temperature, and the doing of a certain 
amount of work AW by the substance. Exactly as in the ideal steam engine as 
described in Art. 109. Therefore, according to equation (56) we have 



Hi ^m 

T +AT ~ T 
also, according to equation (55) we have 

AW = Hi - H2 



(i) 



(ii) 



But it can be easily shown that AW is represented by the shaded area in Fig. 107 
and is equal to (w — i)-Ap. Therefore, using Hi — (w — i)-Ap for H2 in 
equation (i) we get 



Hi 



Hi — ( w — i)'Ap 



T -\- AT 



(iii) 



But Hi is the amount of heat required to melt all but an infinitesimal residue of 
a gram of ice, as stated under 3, and therefore Hi is the latent heat of fusion of ice 



THE SECOND LAW OF THERMODYNAMICS. 



177 



and it is to be represented by the letter L. Therefore, by reducing equation 
(iii), we get 



AT = — (w - i)'Ap 



(iv) 



which is identical to equation (60). 

The value of T at ordinary atmospheric pressure is 273°, the value of L is 
3,360,000,000 ergs per gram (80 calories per gram), the value of w is i. 000127 
cubic centimeter, and the value of i is 1.09 cubic centimeter. Suppose Ap is an 
increase of pressure amounting to one atmosphere, then Ap = 1,013,000 dynes per 
square centimeter. Using these values of T, L, w, i and Ap we get for AT 
a decrement amounting to 0.0075 centigrade degree. That is to say, the freezing 
point of water is lowered to- this extent by increasing the pressure from one atmos- 
phere to two atmospheres, and this is exactly the value obtained by experiment. 



13 



PART III. 
ELECTRICITY AND MAGNETISM. 

Three good books for the beginner are Poynting & Thomson's Electricity and 
Magnetism, Griffen & Co., London, 1904; Hadley's Electricity and Magnetism, 
Macmillan & Co., London, 1910; and Franklin and MacNutt's Advanced Electricity 
and Magnetism, The Macmillan Co., New York, 1915. A very simple discussion 
of the theory of potential (vector analysis) is given in chapter IX of Franklin, 
MacNutt & Charles' Calculus, published by the authors. South Bethlehem, Pa., 
19 13. An important mathematical book which is closely connected to the theory of 
Electricity and Magnetism is Byerly's Fourier's Series and Spherical Harmonics, 
Ginn & Co., Boston, 1895. 

Advanced and special treatises: 

Electricity and Magnetism, Maxwell, 2 vols., 3d edition, Oxford, 1904. 

Theorie der Electrizitdt, Abraham & Foppl, 2 vols., Leipsig, 1904. 

Relativity and the Electron Theory, E. Cunningham, Longmans, Green & Co., 
London, 1915. 

Theory of Electrons, H. A. Lorentz, Columbia University Press, 1909. 

Recent Researches in Electricity and Magnetism, J. J. Thomson, Oxford, 1893. 

Conduction of Electricity through Gases, J. J. Thomson, Cambridge, 1903 and 1906. 

Rays of Positive Electricity, J. J. Thomson, Longmans, Green & Co., London, 
1913- 

Radioactive Substances and their Emanations, Ernest Rutherford, Cambridge, 1913. 
X-rays and Crystal Structure, W. H. & W. L. Bragg, G. Bell & Sons, London, 1915. 
Photo-electricity, H. Stanley Allen, Longmans, Green & Co., London, 1913. 

The Emission of Electricity from Hot Bodies, O. W. Richardson, Longmans & Co., 
London, 1915. 

Electric Waves, Hertz, translated by D. E. Jones, Macmillan & Co., London, 

1893. 
Principles of Electric Wave Telegraphy and Telephony, J. A. Fleming, Longmans, 

Green & Co., London, 1910. 
Wireless Telegraphy, J. Zenneck, translated by A. E. Seelig, McGraw-Hill Book 

Co., New York, 1915. 
Handbook for Wireless Telegraphists, J. C. Hawkhead, revised by H. M. Dowsett, 

Wireless Press Ltd., London, 1915. This book relates chiefly to the apparatus 

and methods of The Marconi Co. 



Magnetic Induction in Iron, J. A. Ewing, Electrician Publishing Co., London, 
1900. 



TABLE. 



Magnetic and Electric Units and Symbols as Used in this Text. 

Centimeter-gram-second system. Ampere-ohm-volt system. 

m = pole strength. Unit pole is 
defined in Art. 128. 

H = field intensity. The gauss is 
defined in Art. 132. 
Note. H is sometimes used 
to represent a quantity of 
heat. 

$ = magnetic flux. The maxwell 
or line is defined in Art. 134. 
i or I = current. The abampere is de- 
fined in Art. 144. 



R = resistance. A portion of a cir- 
cuit has a resistance of one 
abohm when one erg of heat 
is developed in it by one 
abampere in one second. 

E = electromotive force. The ab- 
volt is the electromotive 
force between the terminals 
of a resistance of one abohm 
when a current of one ab- 
ampere is flowing through it. 

L = inductance. A circuit has an 
inductance of one abhenry 
when one abvolt causes the 
current in the circuit to in- 
crease at the rate of one ab- 
ampere per second. 
qor Q = electric charge. The ab- 
coulomb is the charge carried 
in one second by one ab- 
ampere. 

C = capacity. A condenser has a 
capacity of one abfarad 
when one abcoulomb of 
charge is drawn out of one 
plate of the condenser and 
pushed into the other plate 
by an electromotive force of 
one abvolt. 



The ampere is defined as one tenth of an 
abampere. The legal definition of 
the ampere is given in Art. 148. 

A portion of a circuit has a resistance of 
one ohm when one joule of heat is de- 
veloped in it by one ampere in one 
second. 

One ohm = 10^ abohms. 

The volt is the electromotive force be- 
tween the terminals of a resistance 
of one ohm when a current of one 
ampere is flowing through it. 
One volt =108 ab volts. 

A circuit has an inductance of one henry 
when one volt causes the current in 
the circuit to increase at the rate of 
one ampere per second. 

One henry = io~^ abhenrys. 

The coulomb is the charge carried in one 
second by one ampere. 

One coulomb =0.1 abcoulomb. 

A condenser has a capacity of one farad 
when one coulomb of charge is drawn 
out of one plate of the condenser and 
pushed into the other plate by an 
electromotive force of one volt. 
One farad = io~^ abfarads. 



180 



CHAPTER X. 

EFFECTS OF THE ELECTRIC CURRENT. 

1 1 8. The electromagnet. — Figure io8 shows an electric bat- 
tery* connected to a winding of insulated wire on an iron rod. 
When so arranged the iron rod attracts other pieces of iron, and it 
is said to be magnetized. When the wire is disconnected from the 



Wire 



carbon 



j^zinc terminal 




^m 




iron rod 
dry battery '»«»'' 



Fig. io8. 

battery the iron rod loses its magnetism. The iron rod with 
its winding of insulated wire is called an electromagnet. 

A rod of hardened steel may be magnetized in the same way, 
but a rod of hardened steel retains, more or less permanently, a 
large part of its magnetism when the battery is disconnected. 
Such a magnetized rod of hardened steel is called a permanent 
magnet. 

The form of electromagnet which is used In electric bells and 
telegraph instruments is shown in Fig. 109. W^hen the winding of 
wire is connected to a battery the soft iron core becomes a magnet 
and attracts the loose bar of iron. The loose bar of iron is 
sometimes called the armature. 

* This word is here used in its familiar every-day sense. 

181 



1 82 



ELECTRICITY AND MAGNETISM. 



119. The electric lamp. — Figure no shows the essential parts 
of the familiar electric flash-lamp. It consists of a battery and an 
electric lamp connected as shown in the figure. The lamp is a 



.^/Tl 



^^-1 



n 

, 1 




m 
1 

III 1 





iron parts 



side view 



end view 



Fig. 109. 



piece of very fine tungsten wire mounted in an exhausted glass 
bulb with connecting lead-wires of platinum passing through the 
glass. 

When the push-button is pressed the fine wire in the lamp is 
heated to a high temperature and gives off light. 

tampj 



wire 



..* %] § 



wire 




push 
button 



dry battery 



Fig. no. 

120. Electro-plating. — Figure in shows a battery connected 
to two strips of copper C and T both of which dip into a solution 
of copper sulphate. Under these conditions a layer of metallic 
copper is deposited on the metal strip C. This action is called 

electroplating. 

121. The electric current and the electric circuit. — When the 
above described effects are produced an electric current is said to 
flow through the wire. 



EFFECTS OF THE ELECTRIC CURRENT. 



183 



The production of an electric current always requires an 
electric generator such as a battery or dynamo. The path of the 



wire 



carbon 




^ 

zinc terminal 
wire 



cathode- 



C- - T 



-anode 



battery 



copper sulphate solution 



Fig. III. 



current is usually a wire, and this path is called the electric 
circuit. A steady electric current always* flows through a com- 
plete circuit, that is to say, through a circuit which goes out from 
one terminal of a battery (or dynamo) and returns to the other 
terminal of the battery (or dynamo) without a break. Such a 
circuit is called a closed circuit. When the circuit is not complete 
it is said to be an open circuit. The electric current ceases to 
flow through a circuit when the circuit Is opened or broken. 

The electric current has three important effects, namely, the 
magnetic effect which is described in Art. 118, the heating effect 
which is described in Art. 119, and the chemical effect which is 
described in Art. 120. These effects are by no means fully de- 
scribed in Arts. 118, 119 and 120, indeed nearly the whole of the 
elementary study of electricity and magnetism is devoted to 
these three effects. 

122. The electric bell. — The familiar electric bell consists of an 
electromagnet which attracts a piece of iron attached to a small 
hammer, and this hammer is thus made to strike a bell. An 
interesting and important detail of the ordniary electric bell is 

* An electric current which lasts for a very short time, a thousandth of a second, 
for example, can flow in an incomplete or open circuit. In such a case very im- 
portant effects are produced at the place where the circuit is broken. See Art. 199. 



1 84 



ELECTRICITY AND MAGNETISM. 



the arrangement, called an interrupter, for repeatedly making and 
breaking the electrical circuit so as to cause the bell-hammer to 
vibrate continuously. The details of the interrupter are shown 
in Fig. 112. When the current flows, the armature of the electro- 



push button 





dry battery 



Fig. 112. 



magnet is attracted and the circuit is broken at p. The electro- 
magnet then loses its magnetism and the armature is pulled back 
by a spring, thus again closing the circuit at p. This operation 
is repeated over and over again. 

123. Conductors and insulators. — The carbon plate of the 
battery forms a portion of the electrical circuit in Figs. 108, no 
and III, and the solution of copper sulphate forms a portion of 
the electrical circuit in Fig. in. Any substance which can form 
a portion of an electrical circuit, that is, any substance through 
which the "electric current" can "flow" readily, is called an 
electrical conductor. Thus metals, carbon, and salt solutions are 
electrical conductors. Many substances, such as glass, rubber, 
dry wood and air, cannot* form a portion of an electrical circuit 
at ordinary temperatures, that is to say, the electric current 

* This statement is not strictly true; what is called an insulator is merely an 
extremely poor conductor. 



EFFECTS OF THE ELECTRIC CURRENT. 



185 



cannot flow through such substances to any appreciable extent. 
Such substances are called insulators. 

An ordinary telegraph or telephone wire is insulated by being 
supported by glass or porcelain knobs which are called ''insula- 
tors.'" The electric current cannot escape from the wire but it 
must flow along the wire to a distant city and back through 
another wire or through the ground. 

If one were to wind an electromagnet with bare wire the electric 
current would not follow the wire round and round the iron rod, 
and the iron rod would not be magnetized. Therefore the w4re 
which is wound on an electromagnet is insulated by a covering 
of silk or cotton or enamel. 



/O 



CHAPTER XL 



THE MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 

124. The magnetic compass.* — The compass is a magnetized 
needle of hardened steel mounted on a pivot and playing over a 
horizontal divided circle. The direction in which the compass 
needle points at different places on the earth is shown in Fig. 113. 

Everywhere on the heavy Hnes which are marked with a zero in Fig. 113, the 
compass needle points due north and south. In the extreme eastern portions of 
the United States, in western Europe, and over the whole of the North Atlantic 
Ocean the compass needle points to the west of north. Thus everywhere along 
the lines marked lo the compass needle points 10 degrees west of north, everywhere 
along the lines marked 20 the compass needle points 20 degrees west of north, and 
so on. Throughout the western portions of the United States and over the greater 
portion of the North Pacific Ocean the compass needle points to the east of north. 




Fig. 113. 
Lines of equal magnetic declination. 

The deviation of the compass needle to the east or west of north is called the 
declination of the needle. The direction in which the compass needle points at 

* A detailed consideration of the magnetic effects of the electric current depends 
upon a preliminary discussion of magnets and of magnetic fields; and this pre- 
liminary discussion, as here given, is carried far enough to serve as a basis for the 
discussion of induced electromotive force in Chapter XIV. 

186 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 187 

a given place on the earth fluctuates during each day, and the average changes 
from year to year. Figure 113 represents the average decHnation for the year 1905. 

125. Poles of a magnet. — The familiar property of a magnet, 
namely, its attraction for iron, is possessed only by certain parts 
of the magnet. These parts of a magnet are called the poles 
of the magnet. For example, the poles of a straight bar-magnet 
are usually at the ends of the bar. Thus Fig. 114 shows the 
appearance of a bar-magnet which has been dipped into iron 
filings. The filings cling chiefly to the ends of the magnet. 

When a bar-magnet is suspended in a horizontal position by a 
fine thread, it places itself approximately north and south like 
a compass needle. The north pointing end of the magnet is 
called its north pole, and the south pointing end of the magnet 
is called its south pole. 

The north poles of two magnets repel each other, the south 
poles of two magnets repel each other, and the north pole of one 




Fig. 114. 

magnet attracts the south pole of another magnet; that is to 
say, like magnetic poles repel each other, and unlike magnetic poles 
attract each other. 

The mutual force action of two magnets is, in general, resolv- 
able into four parts, namely, the forces with which the respective 
poles of one magnet attract or repel the respective poles of the 
other magnet. In the following discussion we consider only the 
force with which one pole of a magnet acts upon one pole of another 
magnet, not the forces with which one complete magnet acts on 
another complete magnet. 

126. Distributed poles and concentrated poles. — The poles of 
a bar magnet are always distributed over considerable portions 
of the bar. This is especially the case with short thick bars. 



1 88 



ELECTRICITY AND MAGNETISM. 



In the case of a long sliiTi bar magnet, however, the poles are 
ordinarily approximately concentrated at the ends of the bar. 
The forces of attraction and repulsion of concentrated magnet 
poles are easily formulated, therefore the following discussion 
applies to ideally concentrated poles at the ends of ideally slim bar 
magnets. 

127. Definition of unit pole. — Consider a large number of 
pairs of magnets a, b, c, d, etc., as shown in Fig. 115, the two 

magnets of each pair being exactly 
alike.* From such a set it would be 
possible to select a pair of magnets 
such that the north pole of one mag- 
net would repel the north pole of the 
other with a force of one dyne when 
they (the two north poles) are one 
centimeter apart; each pole of such a 
pair is called a unit pole That is, a 

unit pole is a pole which will exert a force of one dyne upon another 

unit pole at a distance of one centimeter, 

128. Strength of pole. — Let us choose a slim magnet with unit 
poles, and let us use one of these unit poles as a "test pole." Any 



N N N N N N ' N N 



d 



B S 



S S 



Fig- 115- 
Pairs of exactly similar magnets 



given pole 
strength m. 



ijn dynes 




one centimeter f 



unit test pole 




m dyneei 



Fig. 116. 

given magnet pole is said to have more or less strength according 
as it exerts more or less force on our "test pole" at a given dis- 
tance. And the force m (in dynes) with which the given pole 

* That is, the magnets of each pair are made of identically the same kind of 
steel, subjected to the same kind of heat treatment and magnetized by the same 
means. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 1 89 

attracts or repels (or is attracted or repelled by) the unit test 
pole at a distance of one centimeter is taken as the measure of 
the strength of the given pole. That is, a given pole has m units 
of strength when it will exert a force of m dynes on a unit pole at a 
distance of one centimeter, as indicated in Fig. 116. 

129. Attraction and repulsion of magnet poles. — Unlike poles 
attract and like poles repel each other, as stated in Art. 125, 
When the two attracting or repelling poles are unit poles their 
attraction or repulsion is equal to one dyne when they are one 
centimeter apart, and the attraction 

or repulsion of two poles whose ^_ __--:^ 

respective strengths are wf and \f>^s^S:^^^^^^--~'^'^ 

m" Is equal to m'm" dynes when m's2units "~^ 

the poles are one centimeter apart. ^. 

Fig. 117. 

One may think of each unit of w' as 

exerting a force of one dyne on each unit of m" . Thus If m' = 3 
units and m'' = 2 units, then the force of attraction or repul- 
sion will be six dynes, as Indicated in Fig. 117, where each dotted 
line represents one dyne. 

130. Coulomb's law. Complete expression for the force of 
attraction or repulsion of two magnet poles. — Coulomb dis- 
covered in 1800 that the force of attraction or repulsion of 
tw^o magnet poles is inversely proportional to the square of the 
distance between them. But the force of attraction or repulsion 
of two magnet poles when they are one centimeter apart is 
m'm" dynes as explained in Art. 129. Therefore, according to 



Coulomb's law, the force of attraction or repulsion Is 
dynes when the poles are r centimeters apart. That is: 



m'm" 



yZ 



m'm" 



F=-^- (61) 

in which m' and m" are the respective strengths of two magnet 
poles, r Is their distance apart in centimeters, and F is the force 
in dynes with which the poles attract or repel each other. 



190 



ELECTRICITY AND MAGNETISM. 



Algebraic sign of magnet pole. — It is customary to consider a 
north pole as positive and a south pole as negative. That is, 
m is positive when it expresses the strength of a north pole and 
negative when it expresses the strength of a south pole. There- 
fore, the product m'm" is positive when both poles are north 
poles or when both poles are south poles, and in this case the 
force F in equation (61) is a repulsion. The product m'm'^ is 
negative when one pole is a north pole and the other pole is a 
south pole, and in this case the force F in equation (61) is an 
attraction. Therefore, when F in equation (61) is positive it is 
a repulsion, and when it is negative it is an attraction. 

131. Magnetic figures. The magnetic field. — When iron 
filings are dusted over a pane of glass which is placed over a 
magnet, the filings arrange themselves in regular filaments, as 
shown in Figs. 118 and 119, if the glass is jarred slightly. Figure 




Fig. 118. 

118 shows the filaments of iron filings in the neighborhood of a 
single bar magnet, and Fig. 119 shows the filaments of filings 
between the unlike poles of two large magnets. 

The region surrounding a magnet is called a magnetic field, 
and the filaments of iron filings in Figs. 118 and 119 show the 
trend of what are called the lines of force of the magnetic field. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 19I 



Indeed any region is a magnetic field in which a suspended^ magnetic 
needle (like a compass needle) points in a definite direction, and the 




Fig. 119. 

direction in which the "north pole" of the needle points is called 
the direction of the magnetic field at the place where the needle 
is suspended. 

132. Intensity of a magnetic field at a point. — ^When a magnet 
is placed In a magnetic field a force is exerted on each pole of the 



N 



N 




Fig. 120. 
The arrows show the forces which act upon the poles of the small magnet. 

magnet by the field. Thus the two arrows in Fig. 120 represent 
the forces which are exerted on the poles of a small magnet which 

* The needle is supposed to be suspended at its center of mass or center of gravity. 



192 ELECTRICITY AND MAGNETISM. 

is placed in the magnetic field between two large magnet poles 
(see Fig. 119). 

The force H in dynes which a magnetic field exerts on a ^^unit 
test pole'' is used as a measure of the strength or intensity of the 
field, and this force-per-unit-pole is hereafter spoken of simply as 
the intensity of the field. The unit of magnetic field intensity 
(one dyne-per-unit-pole) is called the gauss. That is to say, a 
magnetic field has an intensity of one gauss when it will exert 
a force of one dyne upon a unit pole. 

Complete expression for the force exerted on a magnet pole 
by a magnetic field. — A magnetic field of which the intensity is 
H gausses exerts a force of H dynes upon a unit pole as above 
explained, and it exerts a force of mH dynes upon a pole of 
which the strength is m units. That is: 

F = mH (62) 

in which F is the force in dynes which is exerted on a pole of 
strength w by a field of intensity H. 

Uniform and non-uniform fields. — A magnetic field is said to 
be uniform when it has everywhere the same direction and the 
same intensity, otherwise the field is said to be non-uniform. 
The earth's magnetic field is sensibly uniform throughout a room. 
The magnetic field surrounding a magnet is non-uniform. The 
magnetic field surrounding an electric wire is non-uniform. 

133. Direction and intensity of the magnetic field surrounding 
an " isolated " magnet pole of strength M. — By an " isolated " 
magnet pole is meant one pole of a very long slim magnet — the 
other pole being so far away as to be negligible in its action. 

The magnetic field in the neighborhood of an isolated north 
pole is everywhere directed away from the pole as shown by 
the radiating straight lines (lines of force, as they are called) in 
Fig. 121. The magnetic field in the neighborhood of an isolated 
south pole is everywhere directed towards the pole as indicated 
in Fig. 122. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. I93 



Consider two magnet poles M and m which are at a distance 

of r centimeters apart as shown in Fig. 123. The force F with 

Mm 
which M repels m is equal to — — , according to Art. 130; but 

the force exerted on m can also be expressed as equal to mH 





Fie. 121. 



Fig. 122. 



where H is the intensity at m of the magnetic field which is due 



Mm 
to M. Therefore mH = —r- , whence we have : 



H = 



M 



(63) 



in which H is the intensity of the magnetic field produced by 
the pole M a.t a, place which is r centimeters from M. 




r centimeters 5 p 




Fig. 123. 

134. Magnetic flux. — A conception of great importance in the 
consideration of induced electromotive force is the conception of 
14 



194 



ELECTRICITY AND MAGNETISM. 



magnetic flux. The lines of force in Fig. 1 18, as visualized by the 
filaments of iron filings, suggest the idea that something "flows 
out of" one end of the magnet, "traverses" the surrounding 
region in smoothly curved paths, and "flows into" the other end 
of the magnet. 

Definition of magnetic flux. — Figure 124 represents a plane 
area of a square centimeters at right angles to a uniform mag- 
netic field of which the intensity is H gausses. The product 
aH is called the magnetic flux across the area. That is 

^ = aH (64) 

where <l> is the magnetic flux across an area of a square centi- 
meters at right angles to a magnetic field of which the intensity 
is H gausses. 

The unit of magnetic flux is the 
flux across an area of one square 
centimeter (a = i) when the area is 
at right angles to a magnetic field of 
which the intensity is one gauss 
(H = i), or it is the flux across n 
square centimeters of area when the 
area is at right angles to a magnetic 
field of which the intensity is i/:^th 
of a gauss. The unit of magnetic 
flux is properly called the maxwell; 
the common usage however is to call the unit of flux a line of force 
for the following reason : 

Consider any magnetic field whatever, and imagine a surface 
or shell BB which is everywhere at right angles to the field 
as shown in Fig. 125. Imagine the surface BB to be divided 
up into small squares such that one unit of magnetic flux crosses 
each square, and imagine lines of force to be drawn so that one 
line of force passes through each square. Then the magnetic 
flux across any other surface or shell A A placed anywhere in the 
fleld will he equal to the number of these lines of force which cross 
AA. 




Fig. 124. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 195 



135. Proposition. — The amount of magnetic flux $ which 
emanates from a north magnet pole is egual to /[tvM, where M 




Fig. 125. 

is the strength of the pole. Imagine a spherical surface of radius r 
drawn with its center at the pole M as shown in Fig. 126. The 
area of this spherical sur- 
face is 47rr^ (neglecting 
the small portion of the 
sphere which falls inside 
of the slim magnet at 
h). The magnetic field 
at the spherical surface 
due to M is everywhere 
at right angles to the 
surface, and its intensity 

at the surface is if = — 

according to Art. 133. 
Therefore, the magnetic 
flux across the spherical 




Fig. 126. 



M 



surface is equal to /iirr^ ^ "2 ^^ 4xif , according to Art. 

134. But the flux across the spherical surface is the flux that 
emanates from the pole. Therefore the flux that emanates from 
the pole is ^ttM. 



196 



ELECTRICITY AND MAGNETISM. 



136. Oersted's experiment. The needle galvanometer. — The 

magnetic effect of the electric current is a complicated thing, 
and one aspect of it is described in Art. 118. The earliest dis- 
covery relating to the magnetic effect of the electric current was 



wire 



torbon 




y^zinc terminal 

< — 



Wire 



dry 
battery 



S-pole 

pit)ot'-^ 



K 



N'pole 
magnetic needle^ 



coil of wire 



Fig. 127. 
The north pointing pole of the needle is pushed towards the reader. 

made by the Danish physicist Oersted in 1819. Holding an 
electric wire above a compass needle, as shoivn in Fig. 12'/, Oersted 
found that the needle was deflected as indicated by the legend under 
the figure. 

The galvanometer, or more correctly speaking, the galvanoscope, 
is an instrument for detecting the presence of an electric current 

in a circuit. Thus the movement 
of the needle in Fig. 127 shows the 
presence of an electric current in 
the wire, and therefore the arrange- 
ment shown in Fig. 127 might be 
called a galvanoscope. A very 
much more sensitive galvanoscope 
may be made, however, by placing 
a compass needle in a winding of 
wire (a coil) as shown in Fig. 128. 
This form of galvanoscope, the needle galvanoscope, was brought 
to a high degree of perfection by Lord Kelvin, but it is now 
seldom used. 




Wk 



Fig. 128. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 197 

137. Direction of current. — It is very convenient to think of an 
electric current as flowing in a definite direction along a wire, 
and it has been agreed to think of a current as flowing OUT of 
the carbon terminal of a battery, through the circuit and INTO 
the zinc terminal of the battery. 

In the discussion of Fig. iii it was stated that copper was 
deposited upon the metal strip C which is connected to the zinc 
terminal of the battery. Therefore, according to the above agree- 
ment, we are to think of the copper as being carried through the 
solution in the direction of flow of the current. 

The direction of flow of a current through a wire (according 
to the above agreement) can be inferred from the direction of 



wire 




north end of 
needle 



top view 

Fig. 129a. 




Fig. 129&, 



deflection of a compass needle, if we keep in mind the facts which 
are represented in Figs. 129a and 1296. The north pole of the 
compass needle starts to go around the wire in the direction 
indicated by the short arrow a in Fig. 129a, and the current in 
the wire flows in the direction, t, in which a nut would travel on a 
right-handed screw if the nut were turned in the direction in which 
he north pole of the compass needle starts to go around the wire as 
shown by the arrow a in Fig. 129&. 



198 



ELECTRICITY AND MAGNETISM. 



When an iron rod is magnetized by the flow of current round 

it, the north pole of the rod is at the end towards which a nut would 
travel {on a right-handed screw) if the nut were turned in the direction 
in which the current flows round the rod as shown in Fig. 130. 



direction of How 
of current 



S'pole 




N'pole 



When it is desired to show an end-view of a wire through which 
current is flowing, the section of the wire is represented by a 
small circle, current flowing towards the reader is represented 
by a dot in the circle, as if one were looking endwise at the point 
of an arrow ; and current flowing away from the reader is repre- 
sented by a cross in the circle, as if one were looking at the 
feathered end of an arrow, as shown in Fig. 131. 

© 

Fig. 131. 
Current flowing away from reader. Current flowing towards reader. 

138. Another aspect of the magnetic effect of the electric 
current. Side push of the magnetic field on an electric wire. — 




Fig. 132. 
The wire AB is pushed away from the reader. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 1 99 



One aspect of the magnetic effect of the electric current is de- 
scribed in Art. ii8; another aspect of this effect is described in 
Art. 136; and still another aspect of the effect is shown in Fig. 
132. A wire AB through which an electric current is flowing is 
stretched across the end of a magnet, and the wire is pushed 
sidewise by the magnet as stated in the legend under the figure. 
If the current is reversed or if the magnet is turned end for end 
the side push on the wire is reversed.* 

The side force on the wire in Fig. 132 is exerted by the magnet, 
and this force is no doubt transmitted by something which 
connects the magnet and the wire together, namely, the magnetic 
lines of force which emanate 
from the magnet. These 
magnetic lines of force are 
indicated by the dotted 
lines in Fig. 132. 

Figure 133 shows a 
straight wire AB placed 
in a narrow air gap between 
two opposite magnet poles. 
The fine lines across the 
gap represent the magnetic 

lines of force in the air gap, and these lines of force push the 
wire sidewise (away from the reader in Fig. 133). 

When an electric wire is placed in a magnetic field at right 
angles to the lines of force of the field, a force is exerted on the 
wire (a side push on the wire) at right angles to the lines of force 
and at right angles to the wire. 

139. The magnetic field surrounding a straight electric wire. — 
Figure 134 is a photograph of the filaments of iron filings on a 
horizontal glass plate, the black circle is a hole through the plate, 
and a straight electric wire passes vertically through this hole. 
The lines of force of the magnetic field which is produced by an 

* Let it be clearly understood that the wire in Fig. 132 is neither attracted nor 
repelled by the magnet. 




The wire AB 



Fig. 133. 

is pushed away from the 
reader. 



200 



ELECTRICITY AND MAGNETISM. 



electric wire encircle the wire, as shown by the filaments of iron 
fiUngs in Fig. 134. 

140. Explanation of the side push exerted upon an electric wire 

by a magnetic field. — Figure 
ij>V'-"v<^>Vs 119 represents the magnetic 
lines of force between two 
opposite magnetic poles, and 
the attraction of the two op- 
posite poles for each other 
may be thought of as due to 
a state of tension in the lines 
of force. That is, the lines 
of force may be thought of as 
stretched rubber-like fila- 
ments leading from pole to 
pole in Fig. 119, and the at- 
traction of the two opposite 
poles may be thought of as the tendency of these stretched 
filaments to shorten. 

Figure 135 shows hovv the magnetic field between the two oppo- 
site poles in Fig. 119 is modified by the presence of an electric 








Fig. 135. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 201 



steel 
magnet 



wire. The glass plate upon which the filings were dusted in Fig. 
135 is horizontal, and the black circle represents a hole in the 
plate through which the vertical electric wire was placed. The 
lines of force from pole to pole pass mostly to one side of the 
wire in Fig. 135, and the wire is pushed sidewise by the tension 
of the lines for force (tendency of the lines of force to shorten) ^ 

141. The moving coil galvanometer. — 
The side push of a magnetic field on 
an electric wire, as shown in Figs. 132 
and 133 and as explained in Art. 140, 
is made use of in the moving coil gal- 
vanometer and in a common form of 
direct-current ammeter. The essential 
features of the mov^ing coil galvanom- 
eter, usually called the D' Arson val 
galvanometer, from its inventor, are 
shown in Fig. 136. An elongated coil 
of fine insulated wire is suspended 
between the poles NN and SS of 
a strong magnet. The suspending 
wires W and W lead current into 
and out of the coil, the side push of 
the magnetic field upon the vertical 
portions, or limbs, of the coil turns the 
coil, and the motion of the coil is indi- 
cated by a spot of light which is 
thrown upon a fixed scale by the 
mirror. 

The most extensively used type of direct-current ammeter is 
essentially like the D'Arsonval galvanometer, except that the 
moving coil is supported by pivots, and the movement of the 
coil is indicated by a pointer which plays over a divided scale. 
The essential features of a direct-current ammeter of this type 
are shown in Figs. 137 and 138. The vertical portions, or limbs, 
of the movable coil play in a narrow gap space between a fiixed 




Fig. 136. 



202 



ELECTRICITY AND MAGNETISM. 



cylinder of soft iron and the soft iron pole-pieces NN and SS, 
Current is led into and out of the moving coil by means of two 

hair-springs, one at each end of 
the pivot-axis, and the side push 
of the magnetic field on the limbs 
of the coil in the gap spaces turns 
the coil and moves the pointer over 
the scale. 

142. The magnetic blow-out. — 

The side push of the magnetic 
field upon the carrier of an elec- 
tric current, as shown in Figs. 
132 and 133, and as explained in 
Art. 140, is made use of in the 
magnetic blow-out. 
When an electric switch is opened the current continues for a 
short time to flow across the opening, forming what is called an 
electric arc, as shown in Fig. 139. This arc melts the contact 




Fig. 137. 



pointer 





iron 
hrtiftfi 


^ N 


M 




ilhiiiiiiiiiiiMiiiiiiiiiiiiiiiiiiiiiiiiiii 


\ w\ 


"I0 

1 
1 


brass 







Fig. 138. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 203 

parts of the switch, and the switch is soon spoiled. This difficulty 
may be obviated to some extent by always opening the switch 
quickly and unhesitatingly, but where the switch is to be opened 
and closed hundreds of times per day, as in the control of a 



^witch blade 




arc 

switch 8ock^ 



Fig. 139. 



end of^ fl , ..,.<, 

>'_ _ n \< switch hlad§ 




arc 

switch socket 



Fig. 140. 

The arc is pushed towards or away 
from the reader. 



street car motor, it is necessary to blow out the arc so as to avoid 
the rapid wear of the switch contacts by fusion. This blowing 
out of the arc is accomplished by a magnet placed as shown in 
Fig. 140. This magnet pushes sidewise on the arc (towards or 
away from the reader in Fig. 140), and this sidewise push on the 
arc lengthens it very quickly and breaks the circuit. 
143. The electric motor (direct-current type). — The side push 




Fig. 141. 



204 



ELECTRICITY AND MAGNETISM. 



of a magnetic field upon an electric wire, as shown in Figs. 132 
and 133, and as explained in Art. 140, is made use of in the electric 
motor. 

Figure 141 shows an iron cylinder AA placed between the 
poles N and 5 of a powerful electromagnet. The air space 
between each magnet pole and the cylinder is called a gap space; 
and each gap space is an intense magnetic field, as indicated by 
the fine lines (lines of force). Figure 142 shows the cylinder with 




Fig. 142. 

straight wires laid upon its surface, and the dots and crosses 
represent electric currents flowing towards the reader and away 
from the reader, respectively, as explained in Fig. 131. Under 
these conditions the magnetic field in the gap spaces (see fine 
lines of force in Fig. 141) pushes sidewise on the wires, and turns 
the cylinder in the direction of the curved arrows in Fig. 142. 




Fig. 143. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 205 



The arrangement in Fig. 142 is called an electric motor. The 
electromagnet NS is called the field magnet, and the magnetizing 
coils MM are called the field coils or field windings. The rotating 
cylinder AA with its winding of wire is called the armature. 




Fig. 144. 

The arrangement of the wires on the armature and the method 
of leading current into and out of them so that the current may 
flow as indicated by the dots and crosses in Fig. 142 can be most 
easily understood by considering the simplest type of armature 



spider. 




slots 




side view 



end view 



Fig. 145. 



winding, namely, the so-called ring-winding, the essential features 
of which are shown in Fig. 143. An iron ring A A is wound 
uniformly with insulated wire as shown, the ends of the wire 
being spliced together and soldered so as to make the winding 



2o6 



ELECTRICITY AND MAGNETISM. 



endless. Imagine the insulation to be removed from the out- 
ward faces of the wire windings on the ring so that two stationary 
metal or carbon blocks {brushes) a and h can make good 
electrical contact with the wires as the ring rotates. Then if 
current is led into the windings through brush a and out through 
brush &, the current will flow towards the reader in the wires 
which lie under the south pole 5, and away from the reader 
in the wires which lie under the north pole N, as shown by the 
dots and crosses in Fig. 142.* 

In practice, short lengths of wire are attached to the various 
turns of wire on the ring and led to copper bars near the axis of 
rotation, as shown In Fig. 144. These copper bars are insulated 
from each other, and sliding contact is made with these copper 
bars as Indicated in Fig. 144, Instead of being made as indicated 
in Fig. 143. The set of insulated copper bars is called the 
commutator. 

The iron body of the armature (the Iron ring in Figs. 143 and 
144) is called the armature core. This core is built up of ring- 




commutator 



armature 

Fig. 146. 



shaped stampings of soft sheet steel which are supported by the 
arms of a spider as shown in Fig. 145. The entire surface of 
the armature core is slotted (slots being parallel to the armature 
shaft as shown in the side view in Fig. 145), and the armature 
wires are laid in these slots. A few, only, of the slots are shown 
in the end view in Fig. 145. Figure 146 shows a side view of the 
completed armature. 

* Let the reader carefully trace the flow of current in Fig. 143. The current 
which enters at brush a divides, and half of the current flows through the windings 
on each side of the armature. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 207 

The machine which is here described as the direct-current 
motor is properly called the direct-current dynamo. It is a 
motor when it receives electric current from some outside source 




Fig. 147. 
Arrangement of ring armature for a 4-pole field magnet. 

and is used to drive a pump, or a lathe, or a trolley car. Exactly 
the same machine, when driven by a steam engine or water 
wheel, can be used as an electric generator to supply current for 




Fig. 148. 
Arrangement of a 4-pole motor. 



208 



ELECTRICITY AND MAGNETISM. 



driving motors or for operating electric lamps. When so used 
the machine is called a dynamo electric generator or simply a 

generator. 

Bipolar dynamos and multipolar dynamos.— Figure 143 shows 
current led into a ring winding at one point (at the brush a) 
and out at another point (at the brush b). In this case current 
flows towards the reader in all of the wires on one side of the 
armature, and away from the reader in all of the wires on the 
other side of the armature, as indicated by the dots and crosses 
in Fig. 142. Under these conditions the field magnet should 
have two poles, a north pole and a south pole, as shown in Figs. 

141 to 144. 

Figure 147 shows current led into a ring winding at two points 
(at brushes a and a) , and out at two points (at brushes b and 
b). In this case current flows towards the reader in all of the 
wires on the portions FP of the armature, and away from the 




Fig. 149. 



reader in all of the wires on the portions QQ of the armature. 
Under these conditions the field magnet should have four poles, 
two north poles and two south poles, as shown in Fig. 148. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 209 



Figure 148 shows a direct-current dynamo with a four-pole field 
magnet with its magnetizing coils MMMM; and Fig. 149 is a 
general view of a six-pole direct-current dynamo. 



The ring armature and the drum arma- 
ture. — The wire on a ring armature passes 
from one end of the armature to the other on 
the outside of the ring and returns through 
the inside of the ring. In the drum armature, 
however, the wire crosses over to the opposite* 
side of the armature and returns on the out- 
side. 

Every wire on the outside of a ring arma- 
ture may be a commutator bar, or may be 
connected to a commutator bar; whereas every 
second conductor on a drum armature may be 
a commutator bar, or may be connected to a 
commutator bar. 

The ring armature is seldom used in modem 
practice. 



\brush\ 

\U. 'l"'l"Ui ii\ 




Showing relation between ring 
and drum armature windings. 



144. Strength of electric current 
magnetically defined. — Consider a 

straight electric wire stretched across a uniform magnetic field, 
the wire being at right angles to the field as shown in Fig. 
151. Let us suppose, for a moment, that the field is of unit 





N 


1 


1 




* I 

1 

1^ 




1 




wire—*- 






^-direction of 
current 








, 


" ■ 




1 ^ 


1 










• 1 
I' — 



Fig. 151. 
Wire is pushed towards reader. 

* This is for a drum armature which is to be used with a two-pole field magnet. 
The relation between the ring and drum windings may be understood with the 
help of Fig. 150 Each wire a on the interior of the ring may be thought of as 
shifted over to the opposite side of the ring at h, as shown. In this case it is 
evident that the conductor at h must not make contact with the lower brush 
because if it did the cross wires c and d would be short-circuiting connections 
from brush to brush. 
15 



210 ELECTRICITY AND MAGNETISM. 

intensity. The force in dynes with which this unit field pushes 
sidewise on one centimeter of the electric wire has been adopted 
as the fundamental measure of the strength of the current in the 
wire. This force-per-unit-length-of-wire-per-unit-field-inten- 
sity is called simply the strength of the current in the wire, and it is 
represented by the letter 7. The force pushing sidewise on I 
centimeters of the wire is // dynes; and if the field intensity is 
H gausses instead of one gauss, then the force is H times as 
great, or IIH dynes. That is 

F = IIH (66) 

in which F is the force in dynes pushing sidewise on / centi- 
meters of wire at right angles to a uniform magnetic field of 
which the intensity is H gausses, and I is the strength of the 
current in the wire. 

Definition of the abampere. — Definition of the ampere. — 
A wire is said to carry a current The ampere is defined as one 
of one abampere when one tenth of an abampere, 
centimeter of the wire is pushed 
sidewise with a force of one 
dyne, when the wire is stretched 
across a magnetic field of which 
the intensity is one gauss, the 
wire being at right angles to the 
field. The current I in equa- 
tion (66) is expressed in ab- 
amperes when F is expressed 
in dynes, / in centimeters and 
H in gausses. The abampere 
is the c.g.s. unit of current. 

The c.g.s. system of electrical units. — In earlier days, the 
resistance of a particular piece of wire would be used as a unit 
of resistance, the electromotive force of a particular voltaic cell 
would be used as a unit of electromotive force, and current 
values were often specified in terms of the deflections of a par- 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 211 

ticular galvanometer. The introduction of a uniform system 
of units was a great improvement on this old procedure, and it 
was brought about chiefly by Weber and Gauss in Germany and 
by Maxwell and Kelvin in England. This uniform system of 
units was based on the units already in use in mechanics, namely, 
the centimeter, the gram and the second. 

The electrical units now almost universally employed, namely, 
the ampere, the volt, the ohm, the coulomb, the farad, and so 
forth, are not the c.g.s. units but convenient multiples or sub- 
multiples of them. The c.g.s. units as a rule have no names, 
therefore it is convenient to call the c.g.s. unit of current the 
abampere, the c.g.s. unit of resistance the abohm, the c.g.s. unit 
of electromotive force the abvolt, the c.g.s. unit of capacity the 
abfarad, and so forth. 

The c.g.s. units here referred to are the so-called "electromagnetic" c.g.s. 
units. The c.g.s. units of the "electrostatic system" are entirely ignored in this 
text. 

145. The intensity of the magnetic field at the center of a circular coil of wire. — 

If we can calculate the force with which a current in a circular coil of wire acts 

on a magnet pole of given strength placed at the center of the circular coil, we 

can derive an expression for the intensity of the magnetic field at the center of 

the coil due to the current in the coil, because the force exerted on the magnet pole 

by the coil of wire must be equal to mh where m is the strength of the pole and 

h is the intensity at the pole of the field due to the coil. 

Consider, therefore, a magnet pole of strength m placed at the center of a 

circular coil as shown in Fig. 152. This pole produces a magnetic field of which 

m 
the intensity at the wire is — , according to Art. 133, and the lines of force of this 

field are at right angles to the wire. Therefore, according to Art. 144, the wire is 

m 
pushed sidewise (towards the reader in Fig. 152) with a force of 2TrrZ XIX—. 

dynes where 2TrrZ is the length of the wire (Z being the number of turns of wire 
in the coil), and / is the strength of the current in the coi' in abamperes. 

Now the force exerted on the coil by m is equal and opposite to the force 
exerted on m by the coil. Therefore, disregarding signs, we have 

mh = 2TrZ XIX — 

or 

h = (67) 

r 

where h is the intensity at the center of the coil of the magnetic field due to a current 



212 



ELECTRICITY AND MAGNETISM. 



of I abamperes in the coil, r is the radius of the coil in centimeters, and Z is the 
number of turns of wire in the coil. 



Ibattery 




slim magnet 




Fig. 152. 

Circular coil of two turns with current / 
flowing in it and with magnet pole m at 
its center. 



Fig. 153. 

Circular coil of two turns with 
small magnet suspended at its 
center. 



&■ section of coil 
I (9 turns) 



suspended magnet 



^^xis^qf^coil 



146. The tangent galvanometer is an instrument for measuring current. It 
was extensively used in earlier days, but it is now important, only, because it 
illustrates in a very simple manner the measurement of electric current by its 

magnetic effect. It consists essen- 
tially of a circular coil of wire at 
the center of which a small mag- 
net is suspended as shown in Fig. 
153. The suspended magnet car- 
ries a pointer which plays over a 
divided circle by means of which 
one may observe the angle <f> 
through which the suspended mag- 
net is turned when a current I is 
sent through the coil. The coil is 
mounted with its plane vertical and 
magnetic north and south as may 
be seen from Fig. 154. 

When no current is flowing 

through the coil the suspended 

magnet ns points in the di- 

(the horizontal component of the earth's magnetic field). 

2TrZI 




•section of coU 

Fig. 154- 
The plane of the paper is a horizontal plane 



rection of H 
When a current of 



/ abamperes flows through the coil the field h 



( = 



according to Art. 145) is produced, and the suspended magnet points in the direc- 



tion of the resultant field R. 



h 
Now tan <l> = Tft' from Fig. 154. 



Therefore, using 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 213 



2TrZI 



for h and solving for /, we have; 



or 



I in abamperes = — - • tan 4> 
27rZ 



I m amperes = — — • tan <^ 
ttZ 



(68) 



(69) 



The use of the tangent galvanometer for the "absolute" measurement of current, 
that is, for the measurement of current directly in terms of the ampere or abampere 
as defined in Art. 144, depends upon the measurement of the horizontal component 
H' of the earth's magnetic field. A method for carrying out this measurement 
was devised by Gauss and it is known as Gauss's method. A description of this 
method may be found in any good laboratory manual.* An understanding of 
Gauss's method and an understanding of the tangent galvanometer taken together 
will give to the student an idea of how current may be measured fundamentally. 
The most accurate method for the absolute measurement of current is by means of 
the electrodynamometer.f 

* See also Franklin and MacNutt's Advanced Electricity and Magnetism, pages 
11-14, The Macmillan Co., New York, 1915. 

t See Gray's Absolute Measurements in Electricity and Magnetism. 



CHAPTER XII. 

CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 

147. The chemical effect of the electric current again con- 
sidered. — A very beautiful experiment showing the deposition 
of a metal by the electric current is as follows : Two strips of lead 
are connected to seven or eight dry cells (in series) and dipped in to 
a solution of lead nitrate,* as shown in Fig. 155. The flow of 
current decomposes the lead nitrate and deposits beautiful 
feather-like crystals of metallic lead on the lead strip which is 
connected to the zinc terminal of the battery. 



wire 



cathode— 



'Zinc termihat 



1L- 



-= bdtterg 




wire ^ 



* — anode 

-I — II — '-lead nitrate, 
solution 



Jr^^ carbon terminal 



Fig- 155. 

This decomposition of a solution by an electric current is 
called electrolysis, and the solution which is decomposed is called 
an electrolyte. Electrolysis is usually carried out in a vessel 
provided with two plates of metal or carbon as shown in Fig. iii. 
Such an arrangement is called an electrolytic cell, and the plates 
of metal or carbon are called the electrodes. Thus Fig. 156 repre- 
sents an electrolytic cell connected to direct-current supply 

* Ordinary sugar of lead (lead acetate), which can be obtained at any drug 
store, may be used instead of lead nitrate. 



214 



CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 21 5 



D C supply 
mains 



rheostat 



V V f^ 



mains. The electrode A at which the current enters the solu- 
tion is called the anode, and the electrode C at which the 
current leaves the solution is called the cathode. 

The chemical action which is produced by the electric current 
in the electrolytic cell of Fig. 155 is as follows: The lead nitrate 
(PbNOs) is separated into two parts, namely, Pb (lead) and NO3 
(nitric acid radical). The lead (Pb) is deposited on the cathode, 
and the nitric acid radical (NO3) is 
set free at the anode where it com- 
bines with the lead of the anode, 
forming a fresh supply of lead nitrate 
which is immediately dissolved in the 
electrolyte. That is to say, lead is 
deposited on the cathode and dissolved 
off the anode. The dissolving of lead 
off the anode may be made visible as 
follows: Allow the current to flow for 
a few minutes until a deposit of lead 
crystals is obtained on one electrode, 
then reverse the battery connections, 
and the previously deposited lead 
crystals are quickly re-dissolved, a 
new deposit of lead crystals being formed on the other electrode. 

The chemical action produced by the electric current in an 
electrolytic cell takes place only in the immediate neighborhood 
of the electrodes. 

148. Measurement of current by electro-deposition of copper.^ — 

The amount of copper deposited on the cathode in Fig. 11 1 
is proportional to the strength of the current and to the time 
during which the current continues to flow. Therefore, if the 
amount of copper deposited by one ampere in one second is 
known, any steady current can be measured by weighing the 
copper it will deposit during a known time. For example, a cur- 
rent which gave a reading of 2.75 " amperes "^ on an inaccurate am- 
meter was found to deposit 4.24 grams of copper in 4,800 seconds 



fl "* 


^ 1 


>- 




>- 

>■ 





Fig. 156. 



2l6 



ELECTRICITY AND MAGNETISM. 



which is at the rate of 0.0008833 gram of copper per second. But 
very careful measurements have shown that one ampere deposits 
0.000328 gram of copper per second. Therefore the true value 
of the current in amperes is found by dividing 0.0008833 by 
0.000328 which gives 2.69 amperes so that the ammeter reading 
was 0.06 ampere too high. 

The international standard ampere. — Very careful measure- 
ments have shown that one ampere deposits 0.001118 gram of 
silver per second from a solution of silver nitrate in water; and, 
inasmuch as it is very difficult to measure a current accurately 
in terms of its magnetic effect so as to get the value of the current 
directly in amperes, the ampere has been legally defined as the 
current which will deposit exactly 0.001118 gram of silver in one 
second from a solution of pure silver nitrate in water. 

149. Another aspect of the chemical effect of the electric 
current. The voltaic cell or electric battery. — When electric 
current flows through an electrolytic cell chemical action is 
produced. For example, Fig. 157 shows a battery forcing current 

through dilute sulphuric 



batteryj 



cathode -^^ 



— anode 



— dilute 
sulphuric acidi 



acid, the electrodes being 
plates of carbon or lead or 
platinum. The sulphuric 
acid (H2SO4) is decomposed 
by the current, being sepa- 
rated into H2 (hydrogen) 
and SO4 (sulphuric acid 
radical) . The hydrogen 
appears at the cathode as 
bubbles of gas and escapes 
from the cell. The acid 
radical (SO4) appears at the anode where it breaks up into SO3 
and O (oxygen). The oxygen appears in the form of bubbles 
and escapes from the cell, and the SO3 combines with the water 
(H2O) in the cell, forming H2SO4. The net result of the chemical 
action in the cell is therefore to decompose water (H2O) inasmuch 




Fig. 157- 



CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 217 



as hydrogen gas and oxygen gas are given off by the cell. Now 
by burning the hydrogen and oxygen heat energy can be ob- 
tained, and therefore it is evident that work must be done 
{by the battery in Fig. 15'/) to decompose the H2O in the elec- 
trolytic cell. 

Usually the chemical action which is produced by the current 
in an electrolytic cell requires the doing of work as above ex- 
plained, that is to say, an electric generator (battery or dynamo) 
must be used to force the electric current through the electrolytic 
cell. In some cases, however, the chemical action which is 
produced by the flow of current through the electrolytic cell is a 
source of energy. In such a case It Is not necessary to use a 
separate electric generator (battery or dynamo) to force electric 
current through the electrolytic cell, for such an electrolytic cell 
can maintain its own current through the electrolyte from elec- 
trode to electrode and through an outside circuit of wire which 
connects the electrodes. Such an electrolytic cell Is called a 
voltaic cell or an electric battery. That is to say, a voltaic cell is 
an electrolytic cell in which the chemical action produced by the 
flow of current is a source of energy. 

150. The simple voltaic cell. — The simplest example of an 
electrolytic cell in which the chemical ac- 
tion produced by the current is a source of 
energy, Is the so-called simple voltaic cell 
which is shown in Fig. 158. It consists of 
a carbon or copper electrode C and a 
clean zinc electrode Z in dilute sulphuric 
acid. The flow of current through this cell 
breaks up the H2SO4 into two parts, namely, 
H2 (hydrogen) and SO4 (sulphuric acid 
radical). The hydrogen appears at the 
carbon electrode and escapes as a gas, and 
the SO4 appears at the zinc electrode where 

it combines with the zinc to form zinc sulphate (ZnS04). The 
combination of the zinc and the SO4 supplies more energy than 




wit& 



ditute 
sulphuric acid 



Fig. 158. 



2l8 ELECTRICITY AND MAGNETISM. 

is required to separate the H2 and SO4. Therefore the chemical 
action which is produced by the flow of current through the 
cell is a source of energy, and the cell itself maintains a flow of 
current. 

151. Voltaic action and local action. — ^Two distinct kinds of 
chemical action take place in a voltaic cell, namely, (a) the 
chemical action which depends upon the flow of current and 
which does not exist when there is no flow of current, and (b) 
the chemical action which is independent of the flow of current 
and which takes place whether the current is flowing or not. 

The chemical action which depends on the flow of current is 
proportional to the current, that is to say, this chemical action 
takes place twice as fast if the current which is delivered by the 
voltaic cell is doubled. This chemical action is essential to the 
operation of the voltaic cell as a generator of current, its energy 
is available for the maintenance of the current which is produced 
by the cell, and it is called voltaic action. 

The chemical action which is independent of the flow of current 
does not help in any wa^^ to maintain the current, it represents a 
waste of materials, and it is called local action. Local action 
takes place more or less in every type of voltaic cell. It may be 
greatly reduced in amount, however, by using pure zinc, and 
especially by coating the zinc with a thin layer of metallic mercury 
(amalgamation) . 

Example of local action. — The zinc plate in the simple voltaic 
cell which is shown in Fig. 158 dissolves in the sulphuric acid 
even when no current is flowing through the cell, zinc sulphate 
and hydrogen are formed, and all of the energy of this reaction 
goes to heat the cell. If the zinc is very pure and if its surface 
is clean this chemical action takes place very slowly, but if the 
zinc is impure the action is usually very rapid. The hydro- 
gen which is liberated during this local action appears at the zinc 
plate. 

Example of voltaic action. — When the circuit in Fig. 158 is 
closed, hydrogen bubbles begin to come off the carbon electrode, 



CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 219 

and zinc sulphate is formed at the zinc electrode. This is voltaic 
action, and it ceases when the circuit is broken. 

The essential and important feature of voltaic action is that 
it is reversed if a current from an outside source is forced back- 
wards through the voltaic cell, provided no material which has 
played a part in the previous voltaic action has been allowed to 
escape from the cell. Thus in the simple voltaic cell, which is 
described in Art. 150, the sulphuric acid (H2SO4) is decomposed, 
zinc sulphate (ZnS04) is formed at the zinc electrode, and hydro- 
gen is liberated at the carbon electrode. If a reversed current is 
forced through this simple cell, the zinc sulphate previously 
formed will be decomposed, metallic zinc will be deposited upon 
the zinc plate, and the sulphuric acid radical (SO4) will be liber- 
ated at the carbon plate, where it will combine with the trace of 
hydrogen which is clinging to the carbon plate and form sulphuric 
acid (H2SO4). In this simple cell, however, the greater part of 
the liberated hydrogen has, of course, escaped, and the reversed 
chemical action due to a reversed current cannot long continue. 

Local action, being independent of the current, is not affected 
by a reversal of the current. 

152. Primary and secondary chemical reactions in the electro- 
lytic cell. — The decomposition of the electrolyte is the direct or 
immediate or primary effect of the flow of current, therefore, the 
decomposition of the electrolyte may be spoken of as the primary 
chemical action in an electrolytic cell. 

When the decomposed parts of the electrolyte appear at the 
electrodes, chemical action usually takes place between these, 
parts and the electrodes or between these parts and the water in 
the solution, and these chemical actions are called the secondary 
chefnical reactions in the electrolytic cell. For example, the 
primary chemical reaction in the simple voltaic cell which is 
shown in Fig. 158 is the decomposition of the sulphuric acid into 
hydrogen (H2) and sulphuric acid radical (SO4) ; and the com- 
biiiation of the acid radical (SO4) with the zinc of the electrode 
is a secondary reaction. 



220 



ELECTRICITY AND MAGNETISM. 



The primary chemical action in an electrolytic cell usually 
represents the doing of work on the cell, and the secondary 
chemical reactions in an electrolytic cell usually represent the 
doing of work hy the cell ; therefore secondary reactions are very 
important in the voltaic cell or electric battery. 

153. The use of oxidizing agents in the voltaic cell. — The com- 
bination of the SO4 with the zinc of the anode is the secondary 
chemical action in the simple voltaic cell which is shown in Fig. 
158. The available energy of the total chemical action which 
takes place in this cell may be greatly increased, however, by 
providing an oxidizing agent in the neighborhood of the carbon 
electrode so that the hydrogen may be oxidized and form water 
(H2O) at the moment of its liberation by the current. The 
energy of this oxidation increases the available energy of the 
chemical action as a whole, and greatly strengthens the cell as a 
generator of electric current. 

154. The chromic acid cell. — ^The Grenet cell is similar to the 
simple voltaic cell, as shown in Fig. 158, except that the electrode 

C is of carbon, and chromic acid is 
added to the electrolyte to furnish 
oxygen for the oxidation of the hy- 
drogen as it is set free at the carbon 
4 -porous electrode. There is, however, a 
very rapid waste of zinc in this 
cell by local action even when the 
zinc is amalgamated, and the cell 
is now seldom used. A modified 
form of the Grenet cell, known as 
the Fuller cell, is shown in Fig. 159. 
In this cell the electrolyte e is 
dilute sulphuric acid, the zinc anode 
Z is contained in a porous earthen- 
ware cup, and the chromic acid is dissolved only in that portion 
of the electrolyte which surrounds the carbon cathode C In 



wire 



wire 




l}l,ta,iwi/iiiii//,iii,i/iinmi,i/!iiii!/,//l(. 

Fig. 159. 
The Fuller cell. 



CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 221 



this cell there is not a rapid waste of zinc by local action, and the 
cell is extensively used. 

155. Open-circuit cells and closed-circuit cells. — A voltaic cell 
which can be left standing unused, but in readiness at any time 
for the delivery of current when its circuit is closed, is called an 
open-jcircuit cell. A cell to be suitable for use as an open-circuit 
cell should above all things be nearly free from local action. The 
cell most extensively used for open-circuit service is the ordinary 
dry cell. 

A voltaic cell which is suitable for delivering a current steadily 
is called a closed-circuit cell. A cell which is extensively used 
for closed-circuit service is the gravity Daniell cell. 

156. The ordinary dry cell. — A sectional view of this cell is 
shown in Fig. 160. The containing vessel is a can made of sheet 
zinc. This can serves as one electrode of the 

cell, and a binding post is soldered to it. The 
zinc can is lined with several thicknesses of 
blotting paper P, and the space between the 
blotting paper and the carbon rod C is 
packed with bits of coke and manganese 
dioxide. The porous contents of the cell are 
then saturated with a solution of ammonium 
chloride (sal ammoniac) , and the cell is sealed 
with asphaltum cement A. The zinc can is 
usually protected by a covering of paste 
board B. The dry cell has been humorously 
defined as a voltaic cell which, being hermet- 
ically sealed, is always wet; whereas the old- 
style wet cell was open to the air and frequently became dry. 

Reputable manufacturers always stamp the date of manu- 
facture on their dry cells, and a purchaser should not accept a 
cell which is much more than one or two months old. The 
condition of a dry cell is most satisfactorily indicated by observing 
the current delivered when the cell is momentarily short circuited* 

* That is to say, the very low resistance ammeter is connected directly to the 
terminals of the cell. 




Fig. 160. 
The dry cell. 



222 



ELECTRICITY AND MAGNETISM. 



through an ammeter. When the cell has been exhausted by 
use or when it has dried out by being kept too long, the short- 
circuit current is greatly reduced in value. An ordinary dry 
cell, when fresh, should give about 25 or 30 amperes on a momen- 
tary short circuit when the cell is at ordinary room temperature. 

157. The gravity Daniell cell. — This cell consists of a copper 
cathode at the bottom of a glass jar and a zinc anode at the top 

as shown in Fig. 161. The electrolyte Is 
mainly a solution of zinc sulphate. Crys- 
tals of copper sulphate are dropped to 
the bottom of the cell and a dense solu- 
tion of copper sulphate surrounds the cop- 
per cathode. 

This cell has a considerable amount of 
local action when it is allowed to stand 
unused, because of the upward diffusion of 
the copper sulphate. The cell is very 
extensively used in telegraphy and for 
operating the "track circuit" relays in automatic railway sig- 
nalling. 




Fig. 161. 
The gravity cell. 



158. The copper-oxide cell. 

— The cathode of this cell is a 

compressed block of copper 

oxide (CuO), the anode is a ^«^ — | — % 

i. 



A B 




s -^-oil 



— —zinc 



-^- 



copper oxide 



—KOB solution 



plate of zinc, and the electro- 
lyte is a solution of caustic 
potash (KOH).* A sectional 
view of the cell is shown in 
Fig. 162. The zinc anode 
consists of two zinc plates 

(connected together), and the (The zinc plates are connected together 

. . and to binding post £.) 

anode 01 copper oxide is held 



Fig. 162. 
The copper-oxide cell. 



between the zinc plates in a frame of metallic copper. 

* Or a solution of caustic soda NaOH. 



The 



CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 223 

copper-oxide cell is sold under a variety of trade names and it 
is used extensively as a closed circuit cell. 

159. The storage cell. — A voltaic cell may be completely re- 
paired after use by forcing a current backwards through the cell, 
if there is no local action in the cell, and if all of the materials 
which take part in the voltaic action remain in the cell. A voltaic 
cell which meets these two conditions is called a storage cell. 
The process of repairing the cell by forcing a current through it 
backwards is called charging, and the use of the cell for the 
delivery of current is called discharging. 

160. The lead storage cell. — The voltaic cell which is most 
extensively used as a storage cell is one in which one electrode is 
lead peroxide (Pb02) , the other electrode is spongy metallic lead 
(Pb) and the electrolyte is dilute sulphuric acid (H2SO4). This 
cell is called the lead storage cell. The lead peroxide and the 
spongy metallic lead are called the active materials of the cell. 
These active materials are porous and brittle, and they are usually 
supported in small grooves or pockets in heavy plates or grids of 
metallic lead. These lead grids serve not only as mechanical 
supports for the active materials, but they serve also to deliver 
current to or receive current from the active materials which 
constitute the real electrodes of the cell. 





Fig. 164. 



Fig. 163. Fig. 165. 

As a lead storage cell is discharged, the active material on both 
electrodes is reduced to lead sulphate PbS04; and when the cell 
is charged, the lead sulphate on one grid is converted back into 



■■■ 



B 



I^BB 



224 ELECTRICITY AND MAGNETISM. 

spongy metallic lead, and the lead sulphate on the other grid is 
converted back into lead peroxide. 

Figure 163 is a general view of a lead storage cell. One of the 
battery grids is shown in Fig. 164, and a sectional view of the grid 
is shown in Fig. 165. The grid consists of a thick lead plate with 
fine grooves cut into its surface, and the active material is packed 
into these grooves.* 

161. Definition of electrochemical equivalent. Chemical cal- 
culations in electrolysis. — The amount of silver deposited per 
second in the operation of silver plating is proportional to the 
strength of the current in amperes, and the amount of silver 
deposited in one second by one ampere is called the electro- 
chemical equivalent of silver; it is equal to 0.001118 gram per 
ampere per second, or 4.025 grams per ampere per hour. 

In the great majority of cases no material is actually deposited 
at either electrode in the electrolytic cell, but chemical action 
is always produced in the immediate neighborhood of the elec- 
trodes, and the amount of chemical action which takes place in 
a given time due to the flow of a given current through the cell 
can be calculated from very simple data. A statement of the 
method employed in this calculation involves a number of 
chemical terms, and these terms are exhibited in the following 
schedules. 

The valencies of various chemical elements, acid radicals and 
so forth are shown by the numbers in the following exhibit. Thus 
one atom of hydrogen combines with one atom of chlorine and 
each has a valency of i, whereas one atom of copper combines 
with two atoms of chlorine in the formation of cupric chloride 

* A further discussion of this subject of batteries is given in FrankUn'vS Electric 
Lighting, The Macmillan Co., New York, 1912. The Edison nickel-iron storage 
cell is discussed on pages 205-209; a discussion of battery costs is given on pages 
208-211; directions for the management and care of the lead storage battery are 
given on pages 21 1-2 14; and the uses of the storage battery are discussed on pages 
214-255. 

The most extensive treatise on the lead storage battery is Lamar Lyndon's 
Storage Battery Engineering, McGraw-Hill Book Co., New York, 191 1. 



CHEMICAL EFFECT OF THEELECTRIC CURRENT. 225 



SO that the valency of cupric copper is 2. The valency of cuprous 
copper, however, is i . The valency of the sulphuric acid radical 
(SO4) is 2. No attempt is made here to give a general definition 
of valency but merely to recall to the student's mind the knowl- 
edge of valency which he has obtained from his study of chem- 
istry. 

Exhibit of Valencies. 



Name 


Hydrochloric 
acid 


Sodium chloride 


Cupric chloride 


Cuprous chloride 


Chemical symbol 
Valency 


H 

I 


CI 

I 


Na 

I 


CI 

I 


Cu 
2 


CI2 
2X1 


Cu 

I 


CI 

I 


Name 


Sulphuric acid 


Sodium sulphate 


Cupric sulphate 


Cuprous sulphate 


Chemical symbol 
Valency 


H2 
2X1 


S04 

2 


Na2 
2X1 


SO4 
2 


Cu 

2 


SO4 
2 


CU2 
2X1 


S04 

2 



Let m be the atomic weight of an element, or the molecular 
weight of an atomic aggregate or group such as the acid radical 
SO4 or such as the base radical NH4 (which occurs in ammonium 
chloride, NH4CI), and let v be the valency of the element or 
aggregate. Then m/v grams of the element or aggregate is called 
a chemical equivalent thereof. The chemical equivalents of a 
few elements and aggregates are shown in the following exhibit. 

Exhibit of chemical equivalents in grams. 



Symbol of substance 

Atomic or molecular weight . . 

Valency 

Chemical equivalent in grams 



H 


Na 


Ag 


CI 


NO3 


SO4 


Cu* 


Cut 


Zn 


I 


23 


108 


35 5 


62 


96 


63.6 


63.6 


65.4 


I 


I 


I 


I 


I 


2 


2 


I 


2 


I 


23 


108 


35-5 


62 


48 


31.8 


63.6 


32.7 



Al 
27 I 

3 
903 



Note. Atomic weights are given only approximately in round numbers for the 
sake of simplicity. 

THE LAWS OF ELECTROLYSIS. 

I. The amount of chemical action which takes place in an 
electrolytic cell is proportional to the current and to the time 
that the current continues to flow, that is to say, the amount of 
chemical action is proportional to the product of the current 

* Cupric copper, that is copper as it exists in ordinary cupric sulphate, CUSO4. 
t Cuprous copper. 

16 



226 ELECTRICITY AND MAGNETISM. 

and the time. This product may be expressed in ampere-seconds 
or in ampere-hours. Thus ten amperes flowing for five hours 
constitutes what is called 50 ampere-hours. 

II. To deposit one electrochemical equivalent of silver, that is 
108 grams of silver, requires 26.82 ampere-hours, and 26.82 ampere- 
hours will liberate one chemical equivalent of any element or 
radical at an electrode in an electrolytic cell. For example: 

26.82 AMPERE-HOURS WILL LIBERATE 

at the anode at the cathode 

62 grams of NO3 from nitric acid or 23 grams of Na from a sohition of 

any nitrate solution. NaOH, or from a solution of any sodium 

salt. 

48 grams of SO4 from sulphuric acid 31.8 grams of Cu from a solution of 

or any sulphate solution. any cupric salt. 

35.5 grams of CI from hydrochloric 63.6 grams of Cu from a solution of 

acid or any chloride solution. any cuprous salt. 

17.01 grams of OH from a solution of 9.03 grams of Al from a solution of 

caustic soda or potash. any aluminum salt. 

etc. etc. 

EXAMPLES OF ELECTROCHEMICAL CALCULATIONS. 

Note. When one of the elements or radicals which take part 
in a chemical reaction Is di-valent the electro-chemical calcula- 
tions are simplified by considering the amount of chemical action 
which is produced by two times 26.82 ampere-hours. When one 
of the elements or radicals is trivalent the calculations are sim- 
plified by considering the amount of chemical action which is pro- 
duced by three times 26.82 ampere-hours. Thus in the follow- 
ing examples di-valent elements and radicals are involved, and 
the chemical action produced by 53.64 ampere-hours is taken as 
the basis of the calculations; 53.64 ampere-hours liberates /^e;^ 
chemical equivalents of material at each electrode. 

Example (a). Let it be required to find how much zinc, how 
much caustic soda, and how much copper oxide are used up in 
the copper-oxide cell by the delivery of 0.5 ampere for 300 hours, 
local action being ignored. The following schedule shows the 



CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 



227 



reactions at anode and at cathode. The character of these re- 
actions is determined by purely chemical studies, and the reactions 
are supposed to be known ; we are here concerned only with the 
question as to how much of each material corresponds to 34 grams 
of hydroxyl (OH) and to 46 grams of Na, and the schedule shows 
these amounts at a glance. That is to say, 79.6 grams of copper 
oxide is reduced to metallic copper at the cathode, 65.4 grams 



wire 




copper oxide 
cathode 



NaOH solution 



zinc 
anode 



80 



grams 
63.6 18 79.6 46 



grams 



53.64 



34 65.4 80 



143.4 



36 



2NaOH-\-Cu=HzO-^CuO-{-2Na^ ^^^2^—^20H-^Zn-\-2NaOH=Na^n0^^2H^Q 



of zinc is dissolved off the anode, and 80 grams of NaOH (from 
the solution) combines with the dissolved zinc to form sodium 
zincate (Na:Zn02) as a result of the flow of 53.64 ampere-hours 
through the cell. The NaOH which is decomposed by the cur- 
rent, namely, (46 + 34) grams, is made up for or compensated 
by the 80 grams of NaOH which is formed by the reaction at the 
cathode. Therefore 53.64 ampere-hours gives a total consump- 
tion of 79.6 grams of CuO, 65.4 grams of Zn, and 80 grams of 
NaOH; and to get the consumption corresponding to 150 ampere- 
hours the above amounts must be multiplied by 150/53.64. 

Example (b). One result of the discharge of a lead storage 
cell is that HcS04 from the solution combines with the active 
jmaterial of the electrodes thus reducing the strength of the solu- 
tion. Let it be required to find how much H:S04 is taken from 
the solution by 53.64 ampere-hours of discharge. The following 



228 



ELECTRICITY AND MAGNETISM. 



schedule shows the reactions at both electrodes. From this 
schedule we see that (96 + 2) grams of H2SO4 is taken from the 
solution by the electrolytic action, and that another 98 grams of 



wire 



36 303 



tead peroxide 
cathode 



grams 
._>\. 

98 




Hs$0^ solution 



239 



53.64 



96 



spongy lead 
anode 



grams 
207 303 



2H,0+PbSO,=H:^0,-^PbO,^2H <^^^^2elou^^^^ 

H2SO4 is taken from the solution by the reaction at the cathode. 
Therefore a total of 196 grams of H2SO4 is taken from the solu- 
tion. Furthermore 36 grams of H2O is produced by the reaction 
at the cathode. Therefore the solution is weakened by the 
taking of H2SO4 from it and by the adding of H2O to it. 



CHAPTER XIII. 



THE HEATING EFFECT OF THE ELECTRIC CURRENT. 



162. The heating effect of the electric current again considered. 
The current-carrying capacities of copper wires. — The heating 
effect of the electric current is very briefly described in Art. 119. 
The lamp filament in Fig. 1 10 is heated to a very high temperature 
by the current. Careful observation shows that every portion 
of an electric circuit is heated more or less by the current. Thus 
the connecting wires in Fig. no are heated to some extent. 

This heating of electric wires is an important matter because 
excessive heating of a wire in a building involves a risk of fire, 
and because even a moderate rise of temperature may cause a 
serious damage to the insulating material with which the wire is 
covered, especially if the insulating material is rubber. The 
greater the current the hotter the wire will become, and the 
accompanying table of "carrying capacities" of copper wires 
gives the following data: 

Column I gives sizes of wires in Brown and Sharpe's gauge. 

Column 2 gives diameters of wires in mils, a mil equals one 
thousandth of an inch. 

Column 3 gives the current in amperes required to cause a 
bare copper wire of the specified size to be heated 50° F. above the 
temperature of the surrounding air, the bare wire being stretched 
across a room in which the air is still. 

Column 4 gives the current in amperes which a rubber covered 
wire under a wooden moulding can carry without becoming hot 
enough to seriously damage the rubber insulation. 

Column 5 gives the current in amperes which a wire under a 
wooden moulding and with other than rubber insulation can 
safely carry without damage to the insulation. 

Columns 4 and 5 give the limiting carrying capacities of copper 

229 



230 



ELECTRICITY AND MAGNETISM. 



wires according to the National Board of Fire Insurance Com- 
panies. 

Table of Carrying Capacities of Copper Wires. 
(From the National Electrical Code). 



Brown and 
Sharpe ^auge 


Diameters 
in mils. 


Amperes to give 

50° F. rise of bare 

wires in still air 


Carrying capacities 
in amperes of wires 
covered with rubber 


Carrying capacities in 

amperes of wires covered 

with non-rubber 

insulation 


l8 


40 


6.0 


3 


5 


i6 


51 


8.5 


6 


8 


14 


64 


12. 1 


12 


16 


12 


81 


17.1 


17 


23 


10 


102 


24-3 


24 


32 


8 


128 


41.5 


33 


46 


6 


162 


58.8 


46 


65 


5 


182 


69.7 


54 


77 


4 


204 


83.3 


65 


92 


3 


229 


98.8 


76 


no 


2 


258 


117.6 


90 


131 


I 


289 


140.0 


107 


156 





325 


169.8 


127 


18S 


00 


365 


201.5 


150 


220 


000 


410 


240.2 


177 


262 


0000 


460 


286.0 


210 


312 




632 


462.0 


330 


500 






776 


631.0 


450 


680 






1,000 


922.0 


650 


1,000 






1,225 


1,250.0 


850 


1,360 






1,414 


1.550.0 


1,050 


1.670 





For insulated aluminum wire the safe carrying capacity is 0.84 
of that given for copper wire with the same kind of insulation. 

It is important to understand that the heating effect of the 
electric current is not the heating of a wire to a definite tempera- 
ture, it is the generation of heat in the wire at a definite rate, so 
many calories per second. A given wire carrying a given current 
always grows hotter and hotter until heat is given off by the 
wire as fast as heat is generated in the wire by the current. 
Therefore the final temperature of an electric wire depends upon 
the surroundings of the wire. Thus if an electric wire is enclosed 
in a narrow air space its temperature may rise very considerably 
before it gives off heat as fast as heat is generated in it by the 
current, whereas the rise of temperature of the same wire in the 
open air would be much less with the same current flowing 
through it. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 231 



163. The idea of resistance. — The fundamental effects of the 
"electric current" are described in Arts. 118, 119 and 120, and the 
reader should understand that the introduction of the term 
"electric current" in Art. 121 is purely a matter of etymology — 
the science of words ; to say that an electric current flows through 
a wire is merely to say that the wire is connected to a battery or 
dynamo, and that the effects described in Arts. 118, 119 and 120, 
are produced. 

However, the term electric current and the idea of flow have 
been adopted because a battery forcing an electric current 
through a circuit of wire is to some extent analogous to a pump 



Wire 




pipe 



pump 
(fan blower) 




pipe 

Fig, 166. 

forcing water through a circuit 
which goes out from the pump 
electrical resistance also comes 
follows : 

A pump forces air or water 
through a circuit of pipe as 
shown in Fig. 166, and the 
work done in driving the pump 
is converted into heat in the 
pipe, because the flow of the 
water or air through the pipe is 
opposed by friction. There- 
fore we may speak of the fric- 
tion or resistance of the pipe. 



battery «'''*« 

Fig. 167. 

of pipe, that is, through a pipe 
and returns to it. The idea of 
from the hydraulic analogy as 

A dynamo (or battery) forces 
an electric current through a 
circuit of wire as shown in Fig. 
167, and the work done in driv- 
ing the dynamo is converted 
into heat in the wire. There- 
fore we think of the "flow" of 
"electricity" through the wire 
as being "opposed" by a kind 
of "friction," and we speak of 
the resistance of the wire. 



232 .ELECTRICITY AND MAGNETISM. 

A portion of an electric circuit is said to have a high resistance 
when a large amount of heat is generated in the portion by a 
given current in a given time; a portion of an electric circuit is 
said to have a low resistance when a small amount of heat is 
generated in the portion by a given current in a given time. 
Thus, a large amount of heat is generated during a given time in 
the lamp filament in Fig. no, whereas a small amount of heat is 
generated in the connecting wires during the same time by the 
same current. Therefore, the lamp filament has a high resistance 
and the connecting wires have a low resistance. 

164. Joule's law. — An important discovery was made by James 
Prescott Joule about 1850 concerning the relation between the 
strength of a current in amperes and the rate at which heat is 
generated by the current. Joule found that the rate of genera- 
tion of heat in a wire is quadrupled when the strength of the 
current is doubled. For example, if a given current causes heat 
to be generated in a given wire at the rate of one joule per second, 
then twice as much current will cause heat to be generated in the 
same wire at the rate of four joules per second. Joule's discovery 
may be stated in general as follows: the rate at which heat is 
generated in a given piece of wire is proportional to the square 
of the current flowing through the wire. {Joule's law.) 

To say that the rate of generation of heat in a given wire (or 
other conductor) is proportional to the square of the current is 
the same thing as to say that the rate of generation of heat is 
equal to the square of the current multiplied by a constant factor, 
a factor which has a certain definite value for the given piece of wire. 
Let us represent this factor (for a given piece of wire) by the 
letter R. Then RP is the rate of generation of heat in the 
wire by a current of I amperes, that is RP is the amount of 
heat generated in the wire per second, and RPt is the total 
amount of heat generated in the wire during / seconds. There- 
fore we may write: 

H = RPt (70) 

where H is the amount of heat generated in a wire during / 



• HEATING EFFECT OF THE ELECTRIC CURRENT. 233 

seconds by a current of / amperes, and i^ is a factor which has 
a certain definite value for the given piece of wire. 

When a large amount of heat is generated in a wire (or other 
conductor) in a given time by a given current, the factor R is 
large in value; when a small amount of heat is generated in a 
wire in a given time by a given current, the factor R is small in 
value. That is, the factor R is large or small in value according 
as the w^ire has a high or low resistance. Indeed, the value of the 
factor R is used as a measure of the resistance of the wire. 

Example. — The amount of heat generated in a certain glow 
lamp in 10 minutes, as found by a calorimeter, is 7143 calories 
(= 30,000 joules), the current flowing through the lamp being 
0.51 ampere. In electrical calculations it is customary to express 
heat in joules, and it is customary to express time in seconds. 
Therefore, from the above data we get H = 30,000 joules, 
I = 0.51 ampere and / = 600 seconds. Substituting these 
values of H, I, and t in equation (70), we find the value of R 
for the given lamp to be 192.2 joules-per-ampere-square-per- 
second; but one joule-per-ampere-square-per-second is called an 
ohm, as explained in the next article. Therefore the resistance 
of the given glow lamp is 192.2 ohms. 

165. Definition of the ohm. — A wire is said to have a resistance 
of one ohm when one joule of heat is generated in it in one second 
by one ampere. 

When the resistance of a wire (or any portion of an electric 
circuit) is expressed in ohms, then equation (70) gives the amount 
of heat generated in the wire in joules, when the current / is 
expressed in amperes and the time t in seconds. 

166. The rheostat. — Figure 168 shows a pump (a centrifugal 
pump like a fan blovver) forcing a stream of water through a 
circuit of pipe, and Fig. 169 shows a battery forcing an electric 
current through a circuit of wire. The valve in Fig. 168 may be 
closed more and more, thus choking the water stream and in- 
creasing the resistance which opposes the flow of the water 
stream. The arm A in Fig. 169 may be moved so as to include 



234 



ELECTRICITY AND MAGNETISM. 



more and more of the wires ww in the electrical circuit, thus 
making the resistance of the circuit greater and greater. 

To increase the resistance of the water circuit in Fig. i68 by 
partly closing the valve, decreases the water stream (amount of 
water flowing per second). To increase the resistance of the 
electrical circuit in Fig. 169 by including more and more of the 
wires ww, decreases the "stream of electricity" (amperes). 
The arrangement in Fig. 169 (the movable arm A, the wires ww 

rheostat JS 



valve 




wire 




a 



battery 



wire 



Fig. 169. 



and the metal contact-points which are indicated by the black 
dots) is called a rheostat. 

167. Dependence of resistance upon the length and size of a 
wire. Definition of resistivity. — The resistance R oi a. wire of 
given material is directly proportional to the length / of the 
wire, and inversely proportional to the sectional area 5 of the 
wire; that is 



R = k- 
s 



(71) 



in which ife is a constant for a given material, and it is called 
the resistivity'^ of the material. The exact meaning of the factor 
k may be made apparent by considering a wire of unit length 
(/ = i) and of unit sectional area (s = 1). In this case R is 

* Sometimes called specific resistance. The reciprocal of the resistivity of a 
substance is called its conductivity. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 235 



numerically equal to k, that is to say, the resistivity of a material 
is numerically equal to the resistance of a wire of that material 
of unit length and unit sectional area. 

Electrical engineers nearly always express lengths of wires In 
feet and sectional areas in circular mils.* If equation (i) Is used 
to calculate the resistance of a wire in ohms when the length I 
of the wire is expressed in feet and the sectional area j in circular 
mils, then the value of k must be the resistance of a wire of the 
given material one foot long and one circular mil in sectional 
area. The following table gives the resistivities of the more 
important substances together with their temperature coefficients 
of resistance. 

TABLE. — Resistivities and Temperature Coefficients. 



Aluminum wire (annealed) at 20° C 

Copper wire (annealed) at 20° C 

Iron wire (pure annealed) at 20° C 

Steel telegraph wire at 20° C 

Steel rails at 20° C 

Mercury at 0° C 

Platinum wire at 0° C 

German-silver wire at 20° C 

Manganin wire (Cu 84, Ni 12, Mn 4) at 20° C. . . 
"la la" metal wire, hard (copper-nickel alloy) 

at 20° C 

"Climax" or "Superior" metal (nickel-steel 

alloy) at 20° C 

Arc-lamp carbon at ordinary room temperature 
Sulphuric acid, 5 per cent, solution at 18° C. . . . 

Ordinary glass at o" C. (density 2.54) 

Ordinary glass at 60° C 

Ordinary glass at 200° C 



27.4 Xio~^ 

17.24X10"'^ 

95 Xio-^ 

150 Xio~^ 

120 Xio""'^ 

943.4 Xio-^ 

89.8 Xio-7 

212 Xio~' 

475 X10-' 

500 Xio~^ 

800 Xio~^ 
0.005 

4.8 ohms 
10^^ ohmst 
irp ohmst 
10* ohmst 



16.S 
10.4 
58.0 

pit 

72t 

540 

I27t 

286 

30ot 
48ot 



+0.0039 

-f o 0040 

+0.0045 

+0.0043t 

+o.oo35t 

+0.00088 

+0.00354 

+0.00025! 



— o.ooooif 

+o.ooo67t 
— o.ooo3t 
—0.0120* 



* Between 18° C. and 19° C. 

t These values differ greatly with different samples. 

a = resistance in ohms of a bar i centimeter long and i square centimeter 
sectional area. 

6 = resistance in ohms of a wire i foot long and o.ooi inch in diameter. 

j8 = temperature coefficient of resistance per degree centigrade (mean value 
between 0° C. and 100° C). 

Near ordinary room temperature the resistance of a manganin wire is very 
nearly independent of temperature. 



* One mil is a thousandth of an inch. One circular mil is the area of a circle 
of which the diameter is one mil. The area of any circle in circular mils is equal to 
the square of the diameter in mils. Thus a wire 100 mils in diameter has a sec- 
tional area of 10,000 circular mils. 



236 



ELECTRICITY AND MAGNETISM. 



i68. Variation of resistance with temperature. — The electrical 
resistance of a conductor which forms a portion of an elec- 
trical circuit varies with temperature. Consider, for example, 
(a) an iron wire, (b) a copper wire, (c) a platinum wire, (d) a 
German-silver wire, (e) a carbon rod, and (/) a column of dilute 



ohms 



?20 




l6o j8q sqq 



^e^rees centigrade 

Fig. 170. 



sulphuric acid, each of which has a resistance of 100 ohms at 

0° C. Then the values of the resistances of (a), (b), (c), (J), 
(e) and (f) at other temperatures, as determined by experiment, 
are shown by the ordinates of the curves in Fig. 170. As shown 
in this figure, iron and copper increase greatly in resistance with 
rise of temperature, and German silver increases only slightly 
in resistance with rise of temperature. But carbon and sulphuric 
acid decrease in resistance with rise of temperature, carbon 
to a very slight extent and dilute sulphuric acid to a very great 
extent. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 237 

For most practical purposes the curves in Fig. 170 may be 
thought of as straight Unes so that: 

Rt = Ro{i + ^t) (72) 

where Ro is the resistance of a wire at 0° C, Rt is the resistance 
of the wire at f C. and jS is a constant for the material of 
which the wire is made. This constant j8 is called the tempera- 
ture coefficient of resistance of the material. The values of ^ for 
various materials are given in the table in Art. 167. 

169. Power required to maintain a current in a circuit in which 
all of the energy reappears in the circuit in the form of heat in 
accordance with Joule^s law. — Work must of course be done in 
forcing an electric current through an electric motor, but all of 
the work so done does not reappear in the motor wires as heat, a 
large portion reappears at the motor pulley and is delivered as 
mechanical energy to the machine which is driven by the motor. 

Work also must be done in forcing an electric current back- 
wards through, an exhausted storage battery (to charge the 
battery), but all of the work so done does not reappear as heat 
in the circuit, a large portion of the work is expended in bringing 
about the chemical action which takes place as the battery is 
charged. 

When a current is maintained in a simple circuit of wire, or in 
a circuit containing glow lamps, all of the work done in main- 
taining the current does reappear in the circuit as heat, and the 
rate at which work is done in maintaining the current is equal to the 
rate at which heat energy appears in the wire. Now heat energy 
is generated by a current of / amperes at the rate of RP joules 
per second in a circuit of which the resistance is R ohms. 
Therefore, to maintain a current of / amperes in a circuit having 
a resistance of R ohms work must be done at the rate of RP 
joules per second. That is: 

P = RP (73) 

where P is the power in watts (or joules per second) required to 



238 



ELECTRICITY AND MAGNETISM. 



maintain a current of / amperes in a circuit of which the re- 
sistance is R ohms. Equation (73) is true only when all of the 
work expended in maintaining the current reappears in the cir- 
cuit as heat in accordance with Joule's law. 

Example. — A certain electric glow lamp has a resistance when 
hot* of 192.2 ohms. To calculate the power required to maintain 
a current of 0.51 ampere through the lamp, we multiply 192.2 
ohms by (0.51 ampere)^ which gives 50 watts. 

170. Electricity or energy; which? — When water is pumped 
through a pipe it is usually the-amount-of-water-delivered-in-a- 
given-time that is important. The amount of power represented 
by the stream of water is of no great importance. It is the water 
itself that is useful, and the power expended in driving the pump 
is merely enough to carry the water where it is needed. But one 
might conceivably use a pump to drive water through a circuit 
of pipe for the sake of the heating effect of the moving water 
in the pipe or to drive a water motor placed anywhere in the 
circuit of pipe. In such a case one would be interested primarily 
in the amount of power represented by the stream of v/ater 
because the desired effect (heating or motor driving) would 
depend upon the amount of power. 

So it is in the case of the electric current. It is not "elec- 
tricity" (whatever that is) that one uses, it is work or energy; 

and the important thing about 
belt an electric generator (a bat- 

tery or dynamo) is the 
amount of power represented 
by the electric current deliv- 
ered by the machine. This 
may be illustrated most point- 
edly as follows: A wheel A 
drives another wheel -B by a belt, as shown in Fig. 171. A per- 
son knowing nothing at all about machinery and especially a 

* The resistance of a glow lamp changes greatly with the temperature of the 
filament. 




'driven 
wheel 



Fig. 171. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 239 

person having no available words to use in describing such an ar- 
rangement, might look at the continuous stream of leather given 
off by wheel A at the point p and decide to call wheel A a 
leather generator! Everyone knows, however, that a driving 
wheel does not generate leather; it gives off energy or w^ork, and 
the work is transmitted to the driven wheel by the belt. It seems 
very ridiculous to speak of a belt-wheel as a generator of leather, 
and indeed it is equally absurd to speak of a battery or dynamo 
as a generator of elcetricity. One must be careful not to take 
electrical terms and phrases too literally. 

To speak of a dynamo as an electric generator is the accepted 
mode of expression, and it is not, therefore, seriously objection- 
able; but to speak of "electricity" as a motive power indicates a 
very serious misunderstanding. When it is proposed to drive 
a machine by a leather belt it is always understood that some- 
thing must drive the belt, but when it is proposed to drive a ma- 
chine by "electricity" it is not always understood that some- 
thing must drive the "electricity." Electricity as applied in 
the arts is merely a go-between like a leather belt, and no one 
ever thinks of leather as a motive power! 

Electricity! What is electricity? It is what we think of as 
flowing through a wire which is connected to a battery. It is 
what we think of in devising words to describe the effects which 
are represented in Figs. 108, no and iii of Chapter X! What is 
electricity? It is the quintessence of the vocabulary of the 
telephone engineer, the wireless telegrapher and the dynamo 
tender. Electricity belongs to etymology, the Great Science of 
Words ! Let us turn back to the dynamo and consider what it is 
and what it does !* 

171. Electromotive force.— We think of an electric generator 
(battery or dynamo) as a kind of "pump" forcing a "current of 
electricity" through a circuit, the "flow" of electric current 
being opposed by a kind of "resistance." The centrifugal pump, 

* The question "WTiat is electricity?" becomes to some extent legitimate in the 
application of the atomic theory as exemplified in the development of the electron 
theory. 



240 ELECTRICITY AND MAGNETISM. 

or fan blower, is more nearly like the electric generator (battery 
or dynamo) than the ordinary piston pump, and therefore the 
centrifugal pump is used as the basis of the following discussion. 

Figure i68 shows a centrif- Figure 169 shows a battery 

ugal pump forcing a current of forcing a current of electricity 

air or water through a circuit through a circuit of wire, 
of pipe. 

The pump in Fig. 168 exerts The battery in Fig. 169 

a propelling force on the air or exerts a kind of propelling force 

water, thus causing the air or which causes a current of elec- 

water to flow round the circuit tricity to flow round the circuit 

of pipe in spite of the friction of wire in spite of an opposing 

which opposes the flow. resistance, the resistance of the 

wire. 

The propelling force of the The propelling force of the 
pump in Fig. 168 is the pres- battery in Fig. 169 is the "elec- 
sure-difference (in '^pounds" per trical pressure difference" (ex- 
square inch) between the inlet pressed in volts, as we shall see) 
and the outlet of the pump, between the terminals of the 
That is to say, the water or air battery. That is to say, the 
enters the pump at low pres- electric current enters the bat- 
sure, the action of the pump is tery at low electrical pressure, 
to raise the pressure, and the the action of the battery is to 
water leaves the pump at in- raise the pressure, and the cur- 
creased pressure. rent flows out of the carbon 

terminal of the battery at 
increased pressure. 

Throughout this treatise the "propelling force" or "electrical 
pressure-difference" developed by a battery or dynamo is called 
electromotive force or voltage. 

In order to get a better idea as to the electromotive force of a 
battery or dynamo let us consider the familiar gravity cell which 
is described in Art. 157. The chemical action in the cell develops 



HEATING EFFECT OF THE ELECTRIC CURRENT. 241 

energy twice as fast if the value of the electric current which 
flows through the cell (and through the circuit to which the cell 
is connected) is doubled. This is true because the chemical 
action in the cell (the voltaic action) depends upon the current 
as explained in Art. 151, and this chemical action takes place 
twice as fast when the current through the cell is doubled. 

A definite amount of zinc is consumed (by voltaic action) when 
one ampere flows through the cell for one second. Let E be 
the energy in joules developed by the consumption of this amount 
of zinc. That is, E joules per second is the rate at which energy 
is developed by the chemxical action produced by one ampere, 
and EI joules per second is the rate at which energy is developed 
by the chemical action produced by I amperes. Let us assume 
that all of this energy is available for pushing the current throtigh 
the circuit, then the power developed by the battery cell in 
pushing electric current through the circuit will be EI joules 
per second, or EI watts. That is: 

P = EI (74) 

in which P is the power in watts delivered by the gravity cell, 
I is the current in amperes flowing through the gravity cell and 
through the circuit to which the cell is connected, and £ is a 
factor which has a definite value for the given type of cell as 
above explained. This factor E is called the electromotive 
force of the cell. The electromotive force of any generator 
(battery or dynamo) is the factor by which the current I must 
be multiplied to give the power output P of the generator. 

Note. — It may seem from equation (73) that the power P 
delivered to a circuit should be proportional to the square of the 
current. But to increase the current delivered by a given battery 
the resistance of the circuit must be decreased as explained in 
Art. 166. In fact, as we shall see, it is necessary to halve the 
resistance of the circuit in order to double the current. 

172. Ohm's law. — The rate at which energy is delivered by a 
battery is EI watts, as explained above, and the rate at which 
17 















E -- 


= RI 


or, 


solving 


for 


/, 


we 


have: 


I 


E 
~ R 



242 ELECTRICITY AND MAGNETISM. 

heat is produced in the circuit is RP watts according to Art. 164. 
Therefore, if all the energy supplied by the battery is converted 
into heat in the circuit in accordance with Joule *s law, then the 
power developed by the battery must be equal to the rate at 
which heat is generated in the circuit, that is, EI must be equal 
to RP, or cancelling I, we must have: 

(75) 



(76) 



These two relations were discovered by G. S. Ohm in 1827, 
and they constitute what is known as Ohm's law. 

Ohm's law is true when all of the energy delivered by an 
electric generator is used to heat the circuit, that is when 
EI = RI^. Ohm's law is not true when a portion of the energy 
delivered by the generator is used to drive a motor or to produce 
chemical action as in the charging of a storage battery. 

173. Polarization of a battery. — If the electromotive force of a 
battery were invariable, then the current delivered by the battery 
would be doubled by reducing the resistance of the entire circuit* 
to one half, according to Ohm's law. The current delivered by 
a battery is not doubled, however, when the resistance of the 
circuit (the entire circuit) is halved, because the electromotive 
force of a battery falls off more or less with continued flow of 
current, or when the flow of current is greatly increased. When 
a battery delivers current the chemical action quickly exhausts 
the electrolyte (the acid or salt solution) in the immediate neigh- 
borhood of the electrodes (carbon and zinc plates), the energy of 
the chemical action is reduced, and the battery is weakened. 
This weakening shows itself as a decrease of electromotive force, 
and it is called polarization, 

* Including the circuit of wire and the electrodes and electrolyte in the battery 
itself. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 243 



unre 

JT 



current I 



"ii=~ battery 



I 

\E 

I 



C^kunp 



wire 



Fig. 172. 



The gravity Daniell cell does not polarize to any considerable 
extent. The ordinary dry cell polarizes greatly. 

174. Ohm's law and Joule's law are nearly always applied to a 
portion of an electric circuit, not to an entire electric circuit. — 
Consider the electric lamp in Fig. 172. Let R be the resistance 
of the lamp in ohms, and let 

I be the current flowing in 
the circuit in amperes. Then 

RP is the rate in watts at 
which heat is generated in the 
lamp. 

RI {= E) is the electro- 
motive force between the ter- 
minals of the lamp. 

EI ( = RP) is the rate at which energy is delivered to the lamp. 

These statements all refer to the lamp in Fig. 172, not to the 
entire circuit. 

To avoid confusion one should always speak of the current IN 
a circuit; of the resistance OF a circuit (or the resistance of a 
portion of the circuit) ; and of the electromotive force BETWEEN 
THE TERMINALS OF any portion of a circuit. 

Example of the application of Ohm's law. — A lamp or a circuit 
of wire is to be connected to iio-volt mains, and the question 
arises as to how much current will flow through the circuit. 
On the assumption that the voltage between the mains does not 
change, the current in the circuit can be found by dividing the 
voltage by the resistance of the circuit. Thus an ordinary 16- 
candle-power carbon-filament glow lamp has a resistance of 
about 220 ohms when it is hot, therefore if it is connected between 
iio-volt mains, a current of 0.5 ampere will flow through it. 

175. Definition of the volt. — Consider any portion of an electric 
circuit, for example consider the lamp in Fig. 172, and let I be 
the current flowing in the circuit. Then the electromotive force 
E between the terminals of the lamp is equal to RI as stated 
in the previous article, and if E is expressed in ohms and / in 



244 ELECTRICITY AND MAGNETISM. 

amperes, then E {= RI) is expressed in volts. That is, the 
product ohms X amperes gives volts. 

One volt is the electromotive force between the terminals of a 
one-ohm resistance when a current of one ampere is flowing 
through the resistance. 

The electromotive force of an ordinary gravity cell is about i.i 
volts. The electromotive force of an ordinary dry cell is about 
1.5 volts. The voltages commonly used for electric lighting 
and motor service are no volts and 220 volts; that is to say, the 
voltage between the supply wires in a building is usually either 
no volts or 220 volts. The usual voltage for electric railway 
service is 500 volts; that is to say, the voltage between the 
trolley wire and the rails is generally about 500 volts. 

176. The voltmeter. — Consider an ammeter (see Art. 141) of 
which the resistance is R ohms. When a current of / amperes 
flows through the ammeter the electromotive force across the 
terminals of the instrument is RI volts, and the scale of the 
instrument can be numbered so as to give the value of RI in 
volts instead of giving the value of I in amperes. An ammeter 
arranged in this way Is called a voltmeter. 

It would seem from the above that the only difference between 
an ammeter and a voltmeter would be in the numbering of the 
scale; but an instrument which is to be used as an ammeter must 
have a very low resistance in order that it may not obstruct the 
flow of current in the circuit IN which it is connected, and an 
instrument which is to be used as a voltmeter must have a very 
high resistance in order that it may not take too much current 
from the supply mains BETWEEN which it is connected. Thus 
a good ammeter for measuring up to 100 amperes has a resistance 
of about 0.001 ohm so that one-tenth of a volt would be sufficient 
to force the full current of 100 amperes through the instrument. 
A good voltmeter for measuring up to 150 volts has a resistance 
of about 15,000 ohms, so that about o.oi ampere would flow 
through the instrument if it were connected across the terminals 
of a 150-volt generator. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 



245 



177. Measurement of power by ammeter and voltmeter. — The 
power delivered by a battery or dynamo (direct-current dynamo) 
is equal to EI watts, where E is the electromotive force be- 
tween the terminals of the battery or dynamo in volts and / 
is the current in amperes delivered by the battery or dynamo. 
<3^ 




direct -> current 
supply mains 



direct-current 
supply mains 



r 



<E> 




Fig. 173- 

Ammeter and voltmeter connected 
for measuring power output of gener- 
ator. 



Fig. 174. 

Ammeter and voltmeter connected 
for measuring power delivered to L. 



The power delivered to a lamp (or in general to any portion of 
a circuit) is equal to EI watts, where E is the electromotive 
force between the terminals of the lamp in volts, and I is the 
current in amperes flowing through the lamp. 

Figure 173 shows an ammeter and a voltmeter connected so as 
to measure the power delivered by a direct-current generator, 
and Fig. 174 shows an ammeter and a voltmeter connected so 
as to measure the power delivered to a lamp. The voltmeter 
should be momentarily disconnected in Figs. 173 and 174 when 
the ammeter reading is being taken. 

Note. — Power cannot be measured by an ammeter and a volt- 
meter arranged as in Figs. 173 and 174 in an alternating-current 
system. 

178. Voltage drop in a generator. — Let I be the current in 
amperes which is being delivered by a battery (or dynamo), and 
let R be the resistance of the battery in ohms. Then a portion 
of the total electromotive force of the battery is used to force 
the current through the battery itself. The portion so used is 
equal to RI according to Ohm's law. If the total electro- 
motive force of the battery is E volts, then the electromotive 



246 ELECTRICITY AND MAGNETISM. 

force between the terminals of the battery will be {E — RI) 
volts. A voltmeter connected to the battery terminals would 
indicate E volts when the battery is delivering no current,* but 
the voltmeter would indicate {E — RI) volts at the instant the 
battery beginsf to deliver a current of / amperes. The electro- 
motive force RI which is used to overcome the resistance of a 
battery (or dynamo) is called the voltage drop in the battery 
(or dynamo) . 

Voltage drop along a transmission line. — A current of I 
amperes is delivered to a distant motor or to a distant group of 
lamps over a pair of wires, the combined resistance of the pair 
of wires being R ohms. — Let £0 be the voltage across the 
generator, and let Ei be the voltage across the motor or lamps 
as shown in Fig. 175. Then Ei is less than Eo, the difference 
{Eq — EC) is the electromotive force which is used to overcome 
the resistance of both wires, and it is equal to RI volts. This 
loss of electromotive force along a transmission line is called the 
voltage drop along the line. For example, the electromotive force 
across the terminals of a generator is 115 volts. The generator 
supplies 100 amperes of current to a group of lamps at a distance 
of 1,000 feet from the generator, and the wire (2,000 feet of it) 



generator { } fe 





Fig. 175. 

which is used for the transmission line has a total resistance of 
0.05 ohm. Therefore the voltage drop along the line is 100 
amperes X 0.05 ohm, or 5 volts; and the voltage across the 
terminals of the group of lamps is 1 15 volts — 5 volts = 110 volts. 

* Of course the battery delivers current to the voltmeter, but this is a negli- 
gible current because the resistance of the voltmeter is very large as compared with 
the resistance of the battery. 

t Continued flow of current causes a decrease of voltage by polarization as 
explained in Art. 173. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 247 



179. Connections in series. — When two or more portions of an 
electric circuit are so connected that the entire current passes 
through each portion, then the portions are said to be connected 
in series. Thus Fig. 176 shows two lamps, L and L', connected 
in series. The ordinary arc lamps 

which are used to light city streets are 
connected in series, and the entire cur- 
rent delivered by the generator flow^s , 

.r^-battery 
through each lamp ; but the electromotive -^ 

force of the generator is subdivided. 
For example, a generator supplies 6.6 
amperes at 2,000 volts to a circuit con- 
taining 30 arc lamps connected in series. 
The entire current, 6.6 amperes flows 

through each lamp, but the electromotive force across the ter- 
minals of each lamp is 1/30 of 2,000 volts or 67 volts. The 
electromotive force of a generator is subdivided among a number 
of lamps or other units which are connected in series. 

180. The voltmeter multiplying coil. — Given a voltmeter which, 
for example, reads up to 10 volts; one can use such a voltmeter 
for measuring a higher voltage by connecting an auxiliary 
resistance in series with it. Thus Fig. 177 shows a voltmeter V 
of which the resistance is R ohms, and which has an auxiliary re- 



Fig. 176. 

Two lamps in series. 



supply mains 



ytr—R ohms- -^^ 9R ohms H 

I i i 

4 © HvVNAAAAAAAf-J 



U-JRI votts--^=r- — 9RI volts 4 



K-*- ^lORl volts %i 

Fig. 177. 

sistance of gR ohms connected in series with it. Under these 
conditions the voltage between the mains is found by multiplying 



248 ELECTRICITY AND MAGNETISM. 

the reading of the voltmeter by 10. This may be explained as 
follows: Let I be the current flowing through the circuit in 
Fig- 177. Then RI is the electromotive force drop across the 
terminals of the voltmeter, and ()RI is the electromotive force 
drop across the terminals of the auxiliary resistance. Therefore 
RI + ()RI or loi^/ is the electromotive force between the mains; 
but the voltmeter reading gives the value of the electromotive 
force between its terminals, namely RI; therefore the electro- 
motive force between the mains is ten times as great as the 
voltmeter reading. 

181. Connections in parallel. — When two or more portions of 
an electric circuit are so connected that the current divides, part 
of it flowing through each portion, then the portions are said to 
be connected in parallel. Thus Fig. 178 shows two lamps L and 
L' , connected in parallel. The ordinary glow lamps which are 
used for house lighting are connected in parallel between copper 

mains which lead out from the ter- 
I / ""{ minals of the generator; and (if the 

i. 11 resistance of the mains is negligible) 

^battery (T) (E) the full voltage of the generator acts 

T~ f T on each lamp, but the current deliv- 

ered by the generator is subdivided. 
^^^* ^'^^' For example, a iio-volt generator 

Two lamps in parallel. supplies 1,000 amperes tO 2,000 

similar lamps connected in parallel 
with each other between the mains. The full voltage of the 
generator acts on each lamp, but each lamp takes only 1/2000 
of the total current. The current delivered by a generator is 
subdivided among a number of lamps or otlier units which are 
connected in parallel. 

Note. — When a circuit divides into two branches, the branches 
are, of course, in parallel with each other, and either branch is 
called a shunt in its relation to the other branch. 

182. The division of current in two branches of a circuit. — 
Figure 179 shows a battery delivering a current to a circuit which 



HEATING EFFECT OF THE ELECTRIC CURRENT. 



249 



branches at the points A and B. Let / be the current de- 
livered by the battery, /' the current in the upper branch, 
/'' the current in the lower branch, R' the resistance of the 
upper branch, and R^' the resistance of the lower branch. The 
product R'l^ is the electromotive force between the branch 
points A and B, also the product R"I" is the electromotive 
force between the branch points A and B. Therefore we have : 



RT = R'T 



(i) 



The current in the main part of the circuit is equal to the sum 
of the currents in the various branches into which the circuit 
divides. Therefore in the present case we have: 



I = r + r 



(ii) 



By using equations (i) and (ii) the values of J' and /'' can both 
be determined in terms of /, R' and R". 

It is important to note that a definite fractional part of the 
total current flows through each branch; and equation (i) shows 
that the currents r and I'^ are inversely proportional to the 
respective resistances R' and R'\ Thus if R' is nine times 
as large as R'\ then I'' is nine times as large as /'. 




k RY or rY^ )1 

Fig. 179. 



Fig. 180. 
Impracticable arrangement of ammeter 
shunt. 



183. The ammeter multiplying shunt. — ^A low-reading volt- 
meter can be used to measure a higher voltage by connecting 
an auxiUary resistance (a multiplying coil) in series with it as 
explained in Art. 180. A low-reading ammeter can be used to 



250 ELECTRICITY AND MAGNETISM. 

measure a larger current by connecting an auxiliary low resistance 
(a multiplying shunt) in parallel with it. 

It is not practicable, however, to use interchangeable shunts 
with a low resistance instrument (an ammeter). This may be 
illustrated by an example as follows: The ammeter in Fig. i8o 
has, let us say, a resistance of o.oi ohm, and let us suppose that 
a O.OI -ohm shunt 5 is connected across its terminals. Under 
these conditions one half of the total current flows through 
the ammeter and one half flows through s. Therefore the value 
of the total current is twice the ammeter reading. The diffi- 
culty, however, is that if ^ is detachable there is likely to be 
an appreciable* unknown resistance in the contacts of 5 with 
the two binding posts p and p^ so that 5 may be in fact 
10 or 20 per cent, greater than it is supposed to be. Any circuit 
in which binding-post contacts are to be made must be of fairly 
high resistance if the uncertain resistance at the contacts is to 
be negligible. 

Figure i8i shows an ammeter provided with a permanent 
shunt, j and j being soldered joints. In this case the shunt 5 

may be once for all ad- 

I— ©-n S-— 

J I - iW 



justed by the maker of the 
instrument so that the full 



■1 ■ »»» f \ ^ n ■111 iiivjuA uiiiA^^xii. ow uiio-i. cue iuii 

t-AAAA/N I deflection of the instrument 

may correspond to any de- 

Fig. i8i. . ^ ^ 

sired number of amperes. 

Practicable arrangement of permanent am- 
meter shunt. ^^ f^c^ ^ manufacturer 

usually makes the working 
part, C, of all of his ammeters alike. The only difference be- 
tween an ammeter for large current and an ammeter for small 
current is in the resistance of the shunt s. 

184. Combined resistance of a number of branches of a circuit. 
— (a) The combined resistance of a number of lamps or other 
units connected in series is equal to the sum of the resistances of 
the individual lamps, (b) The combined resistance of a number 

* Appreciable, that is, as compared with o.oi ohm. 



HEATING EFFECT OF THE ELECTRIC CURRENT. 25 1 

of lamps or other units connected in parallel is equal to the 
reciprocal of the sum of the reciprocals of the resistances of 
individual lamps. Proposition (a) is almost self-evident. Prop- 
osition (6) maybe established as follows: Let E be the electro- 
motive force between the points A and B where the circuit 
divides into a number of branches (see Fig. 179). Then, accord- 
ing to Ohm's law, we have: 

/' =1 (i) 

I" = J-, ■ (H) 

I'" = — (Hi) 

where R' , R" and R"^ are the resistances of the respective 
branches, and /', I" and I'" are the currents flowing in the 
respective branches. 

Let / be the total current flowing in the circuit (= /' + I" 
+/"')• The combined resistance of the branches is defined 
as the resistance through which the electromotive force E 
between the branch points would be able to force the total 
current J. That is, the combined resistance is defined by the 
equation : 

1=1 (iv) 

in which R is the combined resistance. Adding equations (i), 
(ii), and (iii), member by member, and substituting EfR for 
r -f- r -f r", we have 



whence 



R R''^ R^r "^ R"r ^^^ 



R = (77) 

II I ^"^ 



R' "^ R'r "^ R"r 



CHAPTER XIV. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 
INDUCED ELECTROMOTIVE FORCE. 

185. Back electromotive force in the windings of an electric 
motor. Induced electromotive force. — Consider a motor arma- 
ture of which the resistance is one ohm. When the armature is 
standing still it takes 10 volts to push 10 amperes through the 
armature, according to Ohm's law as expressed by equation (76). 
In this case all the power delivered to the armature is used in 
heating the wires in accordance with Joule's law. 

To push 10 amperes through the motor when it is running 
requires more than 10 volts, because the motor is doing mechan- 
ical work and the power delivered to the motor must be greater 



supply main 



10 volts 



supply main 




supply main 

1 



I 10 amperes 
)\ flowing through 
^ armature 



100 volts 




10 amperes 
flowing through 
armature 



Fig. 182. 
Armature not running. 



supply main 

Fig. 183. 
Armature running. 



than the power lost in heating the wires. Thus Fig. 183 repre- 
sents 100 volts pushing 10 amperes through the running armature. 
It is harder, as it were, to push current through a running 
motor armature than to push current through the same armature 
while it is standing still. Something besides resistance must 
therefore oppose the flow of current through the running arma- 
ture. In fact a back electromotive force exists in the windings of - 
the running armature. This back electromotive force is pro- 
duced by the sidewise motion of the armature wires as they cut 

252 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 253 

across the lines of force of the magnetic field in the gap spaces as 
shown by the fine lines in Fig. 141. An electromotive force 
produced in this way is called an induced electromotive force. 

186. The fundamental equation of the direct-current dynamo. 
— The equation which expresses the electromotive force which is 
induced in the armature windings of a direct-current dynamo in 
terms of various data as explained below is called the funda- 
mental equation of the dynamo. To derive this equation let us 
use c.g.s. units throughout, let us consider the dynamo as an 
electric generator, and let us think of the armature as rotating 
without energy losses of any kind* such, for example, as friction 
losses. Then the mechanical power required to drive the dynamo 
armature is equal to the electrical power output EI of the armature, 
where E is the electromotive force induced in the armature 
windings and / is the current delivered by the armature. Also 
let us limit the discussion to the simple form of dynamo with a 
bipolar field magnet and a ring-wound armature, as shown in 
Figs. 141-144. 

Let r = radius of armature, measured out to the layer of 

wires. 

L = length of armature core parallel to armature shaft. 

This is also the length of the pole faces parallel 

to the armature shaft. 

« 

b = breadth of each pole face measured along the circum- 
ference of the armature where the wires lie. 

Z = number of wires on outside of armature. These wires 
are straight, they lie parallel to the armature shaft, 
and the length of the portion of each which lies in 
the gap space is L. 

n = speed of armature in revolutions per second. 

H = intensity of magnetic field in gap spaces. We aSvSume 

this field to have the same intensity everywhere in 

the gap spaces, we assume the lines of force to 

* This is not necessary but it avoids tedious qualifying specifications in the , 

discussion. * |j| 



254 ELECTRICITY AND MAGNETISM. 

be radial as shown in Fig. 141, and we ignore the 
fringe of the magnetic field which spreads out 
beyond the edges of the pole faces. 

The total current / is supplied by the coming together at the 
brushes of 7/2 abamperes flowing through the windings on each 
side of the armature. That is, the current in each armature wire 
is 7/2 abamperes. 

The side push of the magnetic field on each armature wire in 

the gap spaces opposes the motion of the armature, and is equal 

to LHII2 dynes according to equation (66) of Art. 144; and the 

number of armature wires in the gap spaces at any time is 

26 

— X Z. Therefore the total tangential drag or force on the 
2Trr 

. LHI 2h ^ ^ 
armature wires is X — X Z dynes, and, according to 

2 27rr ^ ' ' & 

equation (20) of Art. 42, the power P required to drive the 
armature against this dragging force is equal to the product of 
this force and the velocity {iirrn) of the armature wires, that is 

X X Z X 2Trrn. But this power is equal to the power 

2 2Trr 

output EI as above explained. Therefore EI = X — 

^ ^ 2 2irr 

X Z X 2'Krn, or 

E = {LhH)Zn (i) 

But the area of a pole face Lh multiplied by 77 gives the amount 
of magnetic flux $ which enters the armature core from the 
N-pole of the field magnet (and leaves the armature core to 
enter the S-pole of the field magnet), according to equation (64) 
of Art. 134. Therefore equation (i) becomes 

E (in abvolts) = ^Zn (78) 

or, since one volt is equal to 10^ abvolts, we have 

E (In volts) = ^Zn X io~^ (79) 

A much simpler expression for Induced electromotive force can 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 255 



be obtained as follows. Let he, Fig. 184, be a wire stretched 
across a magnetic field as shown, the intensity of the field being 
H gausses. The side force F (in dynes) exerted on the wire by 
the field is equal to IIH dynes where / is the length of the wire 
in centimeters and I is the strength of the current in the wire 
in abamperes. 

Suppose the wire to be moving in the direction of the dotted 
arrow at a velocity of v centimeters per second. Then work 
will be done at the rate of 
Fv ergs per second in mov- 
ing the wire in opposition 
to the force F; but, all 
of the work thus spent re- 
appears as the electrical 
work done by the induced 
electromo ti ve force in main- 
taining the current /. Let 
E be the electromotive 
force in ab volts which is 
induced in the moving wire, 
then EI is the rate at 
which work is done by the 
induced electromotive force in maintaining the current. There- 
fore, we must have 

EI = Fv = lIHv 
whence we get 

E (in abvolts) = IHv (80) 

that is, the electromotive force E in abvolts which is induced in 
the moving wire in Fig. 184 is equal to the product of the length 
I of the wire in centimeters, the intensity, H, of the magnetic 
field in gausses, and the velocity, v, of the wire in centimeters 
per second. 

The electromotive force in abvolts which is induced in the moving wire he 
in Fig. 184 is equal to the rate at which the wire cuts magnetic flux. — Consider 
the sidewise distance Ax moved by the wire he in Fig. 184 during the short time 




Fig. 184. 

Dots represent magnetic lines of force 
which are perpendicular to the plane of the 
paper. 



256 ELECTRICITY AND MAGNETISM. 

interval A^. Then: 

A::c = vM (i) 

The area swept over by the wire during the time interval Af is l-i^^x, and the 
amount of flux crossing this area is A$ = HI -Ax, according to equation (64) of 
Art. 134, or, using the value of Ax from (i), we have: 

A$ = HlvAt (ii) 

and, of course, this is the amount of flux which is "cut " b^T- the wire during the time 
interval A^ Therefore, dividing A$ by At we have the rate at which the wire 
cuts flux [in lines of force (or max\vells) per second], and from equation (ii) we have: 

A* 

— = Hlv (iii) 

At ^ ' 

That is, the rate at which the wire "cuts" flux is equal to Hlv, but Hlv is the 
electromotive force in abvolts induced in the moving wire, according to equation 
(80). Therefore the electromotive force in abvolts induced in the moving wire he in 
Fig. 184 is equal to the number of lines of force (maxwells) cut by the wire per second. 

Expression of induced electromotive force in terms of the rate of change of the 
magnetic flux which passes through a circuit. — Figure 184 shows a wire he sliding 
sidewise at a velocity of v centimeters per second along two rails. The electro- 
motive force induced in the moving wire is Hlv, and this electromotive force acts 
to produce current in the circuit abed; the intensity of the magnetic field in Fig. 
184 being H gausses at right angles to the plane of the paper. 

The area of abed is Ix square centimeters, and the magnetic flux $ which 
passes through the opening abed is found by multiplying H by the area Ix; 
that is: 

* = Hlx (i) 

Now if X changes, $ must change HI times as fast, that is: 

d^ ^^,dx 

Ti = ^'J7 («) 

dx 
But, — is the velocity v at which the wire moves sidewise. Therefore equation 

dt 

(ii) becomes: 

d^ 

- = Hlv (iii) 

Therefore, remembering that Hlv is the electromotive force induced in he, and 
remembering that $ is the magnetic flux through the opening abed, we have 
the following proposition: When the magnetic flux $ through the opening of a 
circuit changes, an electromotive force is induced in the circuit, and this induced electro- 
motive force in abvolts is equal to the rate of change of $ in lines of force {maxwells) 
per second. That is: 

£ = -^ . (81) 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 257 



Experiment shows that this equation is true not only when the change of magnetic 
flux is due to m,otion as in Fig. 184, but also when the change of magnetic flux is due 
to a varying m,agnetic field or to a varying state of magnetism of an iron core as in the 
transformer or induction coil. 

The negative sign is chosen in equation (81) for the following reason: The flux 
$ is thought of as going through the opening of a circuit in the direction of the 



magnetic field. Suppose that <J> is increasing. 



Then — 

dt 



is positive, and the 



induced electromotive force E is in the direction around the circuit in which a left- 
handed screw would have to he turned in order that it might travel in the direction of $. 
If we choose the direction-through-the-opening-of-a-circuit which is to be considered 
as positive, then the positive direction around the circuit is considered to be the 
direction in which a right-handed screw would have to be turned in order that 
the screw might travel in the positive direction through the opening of the' circuit. 



d^ 

Therefore when — - 

dt 



is positive E is negative. 



Equation (81) expresses the electromotive force which is induced in a single turn 
of wire. When a region of changing magnetic flux is encircled by Z turns of wire, 
then the induced electromotive force is multiplied Z times, and equation (81) 
becomes: 



- = -f 



(82) 



d$ 



in which — - represents the rate of change of the flux in lines of force (maxwells) 
dt 

per second, and E is the induced electromotive force in ab volts. 

187. The shunt dynamo and the series dynamo. — The field 
magnet of a direct-current generator is generally magnetized or 
excited by current taken from the machine itself. 

The shunt dynamo. — In one type of direct-current dynamo the 



->A/W\/WWW 
field t__ 




Main 



main 



— shunt Held winding 



Fig. 185. 
Shunt dynamo. 

field winding consists of many turns of comparatively fine wire, 

the winding has a comparatively high resistance, the terminals 
18 



258 



ELECTRICITY AND MAGNETISM. 



of the winding are connected directly to the brushes of the 
machine, and from 2 to 10 per cent, of the permissible* current 
output of the generator flows through the winding and excites 
the field magnet, the remainder of the permissible output being 
available for use in the external circuit. In this case the field 

field rheostat 



^hunt 

field 

winding 




mean 



(armature 



mam 



Fig. 186. 
Shunt dynamo diagram. 

winding and the outside circuit (the receiving circuit) are in 
parallel with each other between the brushes so that the field 
winding is in the relation of a shunt to the receiving circuit. A 
direct-current dynamo with its field windings arranged in this 
way is called a shunt dynamo. Figure 185 shows the arrangement 
of a shunt dynamo, and Fig. 186 is a simple diagram showing 
the connections. 

The field rheostat, as shown in Figs. 185 and 186, is an adjust- 
able resistance in circuit with the field windings. By adjusting 

main 




mam 



— senea field wmdmf^ 



Fig. 187. 

Series dynamo. 

* When a dynamo electric generator delivers an excessive current the machine 
becomes dangerously hot. The largest permissible current is that for which the 
rise of temperature is not sufficiently great to endanger the insulation of the 
machine. In some cases sparking at the commutator limits the output of a dynamo 
electric generator. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 259 



mean 



the field rheostat, more or less current can be made to flow 
through the field windings, thus strengthening or weakening the 
field magnet and increasing or decreasing the electromotive force 
of the machine. 

The series dynamo. — In the series dynamo the field winding 
consists of a few turns of coarse wire, the winding has a low 
resistance, the winding is connected in series with the receiving 
circuit so that the whole current output of the machine flows 
through it, and from 2 to 10 per cent, of the electromotive force 
developed by the machine is 
used to overcome the resis- 
tance of the field winding, 
the remainder being available 
for forcing the current through 
the external circuit (receiving 
circuit). A direct-current 
dynamo with its field wind- 
ings arranged in this way is called a series dynamo. Figure 187 
shows the arrangement of a series dynamo, and Fig. 188 is a 
simple diagram showing the connections. 

The compound dynamo. — Most direct-current generators as 
used in practice have two distinct field windings, namely (a) 



-armature 

^series field winding 




maul 



Fig. 188. 
Series dynamo diagram. 



field rheostat 



mmit 



1 


^U-^ 




.jm 


^ 


main 


llllllllll 

nil III 
1 III 1 II 
1 III 1 II 

1 III ill! 
Illllll II 


» ( 






'^. 




t * 






"^series field winding 




















^ shunt fiftlA ininAinn 








"•- ■■" --— W .^..'••^W.Wff 



\ 



Fig. 189. 
Compound dynamo. 

a shunt winding of fine wire which is connected between the 
terminals of the machine, and (6) a series winding of coarse wire 



26o ELECTRICITY AND MAGNETISM. 

through which the entire current output of the machine flows. 
A direct-current dynamo with its field windings arranged in this 
way is called a compound dynamo. The shunt windings usually 
supply the greater part of the field excitation, and the object of 
the series winding is to give an increase of field excitation with 
increase of current output so as to counteract the tendency of the 
electromotive force of the machine to decrease with increase of current 

Held rheostat 

main 




shunt j^ /^^Sir- armature 
field ^^^ 

winding'^ f ^<-^^^^^ "^«^ winding 

main 



Fig. 190. 
Compound dynamo diagram. 

output. Therefore, when properly designed, the compound gen- 
erator gives a nearly constant voltage however its current output 
may vary. Figure 189 shows the arrangement of the compound 
generator, and Fig. 190 is a simple diagram showing the connec- 
tions. 

188. The starting rheostat for the direct-current motor. — 

When a motor is running, the flow of current through the arma- 
ture windings is opposed only in small part by the resistance of the 
windings; the chief opposition to the flow of current is the back 
electromotive force in the armature windings, as explained in 
Art. 185. For example, a fully loaded motor takes 50 amperes 
from iio-volt mains, and the resistance of the armature windings 
is 0.3 ohm. Multiplying the resistance of the armature windings 
by the current we get 15 volts (0.3 ohm X 50 amperes = 15 
volts), which is the portion of the supply voltage (no volts) 
that is used to overcome the resistance of the armature. The 
remainder, namely 95 volts, is used to overcome the back electro- 
motive force in the armature windings. 

If there were no back electromotive force the supply voltage 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 261 



would produce a current of 1 10 volts -^ 0.3 ohm, or 367 amperes 
through the armature windings, according to Ohm's law. Now, 
as a matter of fact, there is no back electromotive force when the 
motor armature is standing still, and, therefore, if the motor 
were to be started by connecting the armature terminals (the 
brushes) directly to the supply mains, a current of 367 amperes 
would flow through the armature at the beginning. This exces- 
sive flow of current through a motor armature would damage the 
motor, and it is therefore always necessary to connect a rheostat 
in series with a motor armature at starting. As the motor speeds 
up the rheostat resistance may be cut out more and more, and 
the greater and greater back electromotive force due to the 
greater and greater speed keeps the current down to a moderate 
value. 

Figure 191 is a diagram showing how a shunt dynamo is con- 
nected to the supply mains when the dynamo is to be used as a 
motor. When the switch is closed the shunt field winding is at 

^switch 
shunt ^ > ^armature 
held 
winding 




'rheostat arm 

Fig. 191. 

Connection diagram for starting shunt motor. 

once connected across the supply mains as may be seen by in- 
specting the figure. Therefore the field magnet of the motor 
is at once fully excited. The closing of the switch also connects 
the motor armature across the supply mains but in series with 
the rheostat RR. Then, as the motor speeds up, the arm of the 
rheostat is moved so as to cut out resistance slowly, and the 
rheostat arm stands permanently in the dotted position LA 
while the motor is running. 



262 



ELECTRICITY AND MAGNETISM. 



In the starting of a series motor, the field winding, the arma- 
ture, and a starting rheostat, all in series, are connected to the 
supply mains, and resistance is cut out of the rheostat as the 
motor speeds up. 

189. The alternating-current djniamo. — An alternating-current 
dynamo is usually called an alternator. An ideally simple 



moving wife 






TM.^^ metal rings— ^-i^ 
brushes wj fr 



Fig. 192, 



external eireuU 

Fig. 193. 



alternator is shown in Figs. 192 and 193. A wire WW is sup- 
ported by arms mm which are fixed to a rotating shaft as shown 
in Fig. 193, and as the shaft revolves the wire sweeps across the 
pole faces N and S of an electromagnet as shown in Fig. 192. 
The ends of the wire WW are attached to two metal rings r 
and r\ and metal or carbon brushes a and b rub on these 
rings, thus keeping the ends of the moving wire connected to an 
external circuit as shown in Fig. 193. 

While the wire WW is sweeping across the north pole N 
an electromotive force is induced in the wire in one direction; 
and while the wire WW is sweeping across the south pole 5 
an electromotive force is induced in the wire in the opposite 
direction. This rapidly reversed electromotive force is called an 
alternating electromotive force, and it produces an alternating 
current in the moving wire and in the external circuit to which the 
moving wire is connected. 

An improvement on the ideally simple alternator of Fig. 192 
would be to place the moving wire WW in a slot in a rotating 
iron cylinder A A as shown in Fig. 194; because to do so would 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 263 



give a firm support for the moving wire, and the presence of the 
iron cyUnder AA would greatly intensify the magnetic field 
(see lines of force in Fig. 141). The iron cylinder A A is built 
up of thin sheet iron disks or stampings, and it usually has a 
large number of slots in which 
many wires are placed as de- 
scribed later. The electro- 
magnet NS in Figs. 192 and 
194 is called the field magnet 
of the alternator. The iron 
cylinder AA with its slots 
and wires is called the ar- 
mature. The cylinder A A 
itself is called the armature core. The insulated metal rings r 
and r' are called the collector rings. The collector rings are 
usually placed side by side at one end of the armature. 

The field magnet of an alternator is always excited (magnetized) 
by direct current, and this direct current is usually supplied by a 
small auxiliary direct-current generator which is called the exciter. 




Fig. 194. 




Fig. 195- 
Possible armature winding diagram for 
4-pole alternator. 



Fig. 196. 

Possible armature winding diagram for 
4-pole alternator. 



Commercial alternators nearly always have multipolar field 
magnets, whereas Figs. 192 and 194 show bipolar field magnets. 
Also the armature windings always consist of many wires in 



264 



ELECTRICITY AND MAGNETISM. 



many slots. Thus, Figs. 195 and 196 show two possible arrange- 
ments of the armature windings of a 4-pole alternator. The 
dotted circles represent front and back ends of the armature 
core, the short, heavy, radial lines represent the wires which lie 
lengthwise (parallel to the armature shaft) in the armature slots, 
the curved lines F represent the cross connections on the front 
end of the armature, the curved lines B represent the cross 
connections on the back end of the armature, and the straight 
lines C represent the connections to the collector rings (the 
small black circles) upon which the brushes rub. The field 
magnet poles are of course very close to the armature but they 
are shown widely separated in Figs. 195, 196 and 197, so as to 
give room to show the back connections B. 

Figure 197 shows a possible arrangement of the armature 
windings of an 8-pole alternator. 

The simple alternator above described is called a single-phase 
alternator. It has a single armature winding and two collector 

rings. The two-phase alter- 
nator has two distinct arma- 
ture windings, each winding 
being connected to two col- 
lector rings (four collector 
rings in all). The three-phase 
alternator has three distinct 
armature windings, each 
winding being connected to 
two collector rings (six col- 
lector rings in all). Because 
of certain relations between 
the three distinct alternating 
currents which are delivered 
by the three armature wind- 
ings of a three-phase alternator (delivered to three distinct receiv- 
ing circuits, of course) it is possible to use only three collector 
rings; and this is the usual practice. 




Fig. 197. 

Possible armature winding diagram for 
8-pole alternator. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 265 

Definition of cycle. Definition of frequency. — ^The electro- 
motive force developed by a direct-current dynamo is fairly 
steady in value and unchanging in direction, and the current 
which is delivered by such a machine flows steadily in one direc- 
tion through the receiving circuit. But the electromotive force 
of an alternator (and also the current which is delivered by the 
machine to a receiving circuit) is subject to rapid reversals of 
direction. Two successive reversals constitute what is called 
a cycle, and the number of cycles per second is called the frequency 
of the alternating electromotive force or current. The electro- 
motive force of a two-pole alternator like Figs. 192 and 194 is 
reversed twice for each revolution of the armature. But two 
reversals constitute a cycle; therefore the frequency of the elec- 
tromotive force of a two-pole alternator in cycles per second 
is equal to the speed of the armature in revolutions per second. 
A four-pole alternator gives two complete cycles of electromotive 
force during each revolution, and an eight-pole alternator gives 
four complete cycles of electromotive force during each revolution 
of the armature. 

The standard frequencies of alternating electromotive force 
and current in practice are 25 cycles per second (50 reversals per 
second) and 60 cycles per second (120 reversals per second). 

190. Electromotive force induced by increase or decrease of 
magnetism of an iron rod. — The electromotive force induced in 
the wire on a dynamo armature is due to the motion of the wire 
across the magnetic field, or, as it is sometimes stated, the elec- 
tromotive force is due to the ''cutting" of the lines of force of 
the magnetic field in the gap space by the armature wires as they 
move sidewise. An electromotive force is also induced in a winding 
of wire on an iron rod while the magnetism of the iron rod is being 
increased or decreased. Figure 198 shows an iron core CC (made 
of strips of sheet iron) with a winding of wire PP upon it. The 
winding is connected to alternating-current supply mains, and 
the rapid reversals of the alternating current in the coil PP 
cause the iron core CC to be rapidly magnetized and demagne- 



p 


C 

■1 


1 y 

1 n^ 


^""'j^^s*^^ 


p i^pl***'^*'^^**' alternating-current 


■ff-s^^Mj ""Sps**^ in«n« 


r'-.^ 


iii( 


c-^ 



Fig. 198. 



266 ELECTRICITY AND MAGNETISM. 

tized, first in one direction and then in a reversed direction; that 
is to say, the upper end of the core is at one instant a north pole 
and at the next instant a south pole. An auxihary coil, SS 

consisting of a few turns of 
wire, has its terminals TT 
connected by a fine wire w» 
Under these conditions the 
fine wire becomes red hot. 
The heating of the fine wire 
shows the existence of an 
electric current in the wire and 
in the coil 55. This current 
is an alternating current, and it is produced by an alternating 
electromotive force which is induced in the coil 55 by the 
magnetic reversals of the core CC. 

191. The alternating current transformer. — The alternating- 
current transformer is a device essential!}^ like the arrangement 
shown in Fig. 198. The winding FP which receives alternating 
current from some outside source is called the primary coil of the 
transformer, and the coil 55 in which an alternating current is 
produced by the reversals of magnetism of the core is called the 
secondary coil of the transformer. In the commercial transformer 
the iron core forms a closed circuit (a closed "magnetic circuit") 
as shown in Fig. 199. The iron core CC is built up of sheet iron 
stampings, and the two coils P and 5 are wound, one over the 
other. One of the coils P or 5 usually contains many turns of 
fine wire, and the other contains few turns of coarse wire. Either 
coil may be used as the primary coil. 

Step-down transformation. — ^A small alternating cuirent may 
be delivered at high voltage to the coil-of-many-turns, in which 
case the coil-of -few- turns will deliver a large alternating current 
at low voltage. This constitutes what is called step-down trans- 
formation. 

Step-up transformation. — A large alternating current may be 
delivered at low voltage to the coil-of -few- turns, in which case 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 267 




Fig. 199. 

Alternating-current trans- 
former. 



the coU-of-many-turns will deliver a small alternating current 
at high voltage. This constitutes what is called step-up trans- 
formation. 

In practice a transformer like Fig. 199 is 
placed in an iron case which is filled with 
mineral oil. Thus Fig. 200 shows a step- 
down transformer in an iron case mounted 
on a pole. Alternating current is delivered 
to the coil-of-many-turns of the trans- 
former from high-voltage street mains ss, 
and a large alternating current at low volt- 
age is delivered by the coil- of- few- turns 
and led into an adjoining house over the 
house wires hh. The small cases FF con- 
tain safety fuses. In large cities the 
high -voltage street mains are usually laid 
underground, and the step-down trans- 
formers are usually placed in vaults under the street. 

192. High voltage must be used in the long-distance trans- 
mission of power. — In order to appreciate the very great practical 
importance of the alternating-current transformer one must 
understand that high voltage is necessary for long-distance trans- 
mission of power, whereas low voltage is necessary for supplying 
power to lamps and motors. Thus the voltage between the 
street mains 55 in Fig. 200 is usually 1,100 or 2,200 volts, 
and the voltage between the house wires hh is usually no volts. 

Consider the transmission of power by the pumping of water 
through a long pipe to a water motor. A given amount of power 
can be thus transmitted by using a very large pipe to carry a 
large volume of water per second from a low-pressure pump to 
a low-pressure water motor; or the same amount of power can 
be transmitted by a small pipe carrying a small amount of water 
per second from a high-pressure pump to a high-pressure water 
motor. If power were to be transmitted over a considerable 
distance in this way the cost of the pipe would be the most 



268 



ELECTRICITY AND MAGNETISM. 



important item of cost in the entire installation, and it would 
therefore be most economical to use high-pressure water so as to 
be able to use a small pipe. 

Consider the transmission of power by an electric current. A 
given amount of power can be transmitted by using a large wire 

to carry a large current from 
a low- voltage generator to a 
low-voltage motor ; or the 
same amount of power can be 
transmitted by using a small 
wire to carry a small current 
from a high- voltage generator 
to a high-voltage motor. The 
cost of the transmission line 
is one of the largest items of 
expense in an installation for 
the long-distance transmission 
of power by the electric cur- 
rent, and therefore it is most 
economical to transmit the 
power by small current at high 
voltage so as to be able to use 
small wires. 

A difficulty in the use of high 
voltage for the transmission of 
power is that it is a practical necessity to deliver power to motors 
and lamps at low voltage. Therefore, since economical long- 
distance transmission requires the use of a high voltage in order 
to reduce the cost of the transmission line, it is necessary to use 
a device for transforming the power at the receiving station from 
high- vol tage-and-small-current to low-voltage-and-large-current. 
The advantage of the alternating-current system over the 
direct-current system lies almost wholly in the cheapness of con- 
struction and the economy of operation of the alternating-current 
device which is used for this transformation, namely, the alter- 




Fig. 200. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 269 

natlng-current transformer. To accomplish the same trans- 
formation in the direct-current system would require the use of 
a specially constructed motor operated by the high-voltage 
supply, and this motor would have to drive a low- voltage gen- 
erator for delivering current at low voltage to the receiving 
apparatus. Such a combination of motor and generator is called 
a motor- generator , and the advantages of the transformer as 
compared with the motor-generator are as follows: 

A transformer costs only one third or one fourth as much as a 
motor-generator of the same capacity. 

A transformer can be placed anywhere, and it needs only to 
be occasionally inspected ; whereas a motor-generator requires a 
building and the care of an attendant. 

A transformer wastes only two or three per cent, of the power; 
whereas a motor-generator wastes 20 or 30 per cent, of the power. 

The essential parts of an installation for the long-distance 
transmission of power are as follows: A water wheel drives an 
alternator which delivers alternating current at a moderately low 
voltage to a step-up transformer. The step-up transformer 
delivers alternating current to the transmission line at a very 
high voltage. At the- other end of the transmission line the small 
current at high voltage is delivered to a step-down transformer 
which in turn delivers a large current at low voltage to the 
receiving apparatus. If direct current is desired at the receiving 
station, the step-down transformer delivers alternating current 
at low voltage to a machine called a rotary converter,"^ and this 
converter delivers direct current. 

193. The induction coil. — An iron rod or core wound with insu- 
lated wire can be repeatedly magnetized and demagnetized by 
connecting a battery to the winding and repeatedly making 
and breaking the circuit; and the increase and decrease of mag- 
netism of the core thus produced can be utilized to induce elec- 
tromotive forces in an auxiliary coil of wire wound on the iron 

* For a description of the rotary converter see Franklin & Esty's Dynamos and 
Motors, Chapter XIII, The Macmillan Co., 1909. 






270 



ELECTRICITY AND MAGNETISM. 



core. Such an arrangement is called an induction coil. The 
winding through which the magnetizing current from the battery- 
flows is called the primary coil, and the auxiliary winding in which 




Fig. 201. 
Induction coil. 

the desired electromotive forces are induced is called the secondary 
coil. The iron core is always made of a bundle of fine iron wires 
or strips of sheet iron. 

A general view of an induction coil is shown in Fig. 201 , and the 
diagram of connections is shown in Fig. 202'. When the iron core 




battery^ 

Fig. 202. 
Induction coil diagjam. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 271 

is magnetized, the block of iron a is attracted, and the battery 
circuit is broken at the point p. The iron core then looses its 
magnetism, and the spring ^ brings the points a.t p into contact 
again so that the battery current again flows through the circuit 
and magnetizes the iron core. The iron block a is then at- 
tracted again, and the above action is repeated. 

When the iron core of an induction coil is magnetized, a 
momentary pulse of electromotive force is induced in the second- 
ary coil; and when the iron core is demagnetized a reversed 
momentary pulse of electromotive force is induced in the second- 
ary coil. Electromotive forces are induced only while the core 
is being magnetized or demagnetized, and each pulse of electro- 
motive force may be made very large in value (many thousands 
of volts) by using many turns of wire in the secondary coil and 
by providing for the quickest possible magnetization or demagne- 
tization of the core. A battery cannot, however, magnetize a 
core very quickly when connected to a magnetizing coil; in fact 
a very considerable fraction of a second is required for the core 
to become magnetized. Therefore during the magnetization of 
the Iron core of an induction coil the electromotive force induced 
in the secondary coil is a comparatively weak pulse of fairly long 
duration. 

On the other hand the use of the condenser CC, Fig. 202, 
causes the iron core of the induction coil to be demagnetized very 
quickly as explained in Art. 198, and this quick demagnetization 
induces in the secondary coil an intense pulse of electromotive 
force of very short duration. 

The iron core of the alternating-current transformer may be as shown in Fig. 
198, but it is better if the iron core forms a complete "magnetic circuit" as shown 
in Fig. 199. The induction coil must have its iron core in the form of an open 
"magnetic circuit" as shown in Fig. 202, because after the core has been magnetized 
it is the energy of this magnetism which becomes available when the primary circuit is 
broken, and the greater part of the available . energy of a magnet resides in the 
magnetic field near the poles of the magnet. 

194. The telephone. — ^The telephone set includes a transmitter, 
a receiver and an arrangement for calling. The transmitter is a 
device for producing over the line a current which is reversed 



272 



ELECTRICITY AND MAGNETISM. 




Fig. 203. 
Telephone transmitter. 



with each to and fro movement of a diaphragm, the diaphragm 
being set into vibration by a speaker's voice; and the receiver 
is a device in which a diaphragm is set into vibration by these 

rapidly reversed currents 
(which come to it over the 
line from the transmitter) 
thus reproducing the orig- 
inal sound. 

The transmitter. — A 
sectional view of a tele- 
phone transmitter is 
shown in Fig. 203. It 
consists of a small quan- 
tity of finely granulated 
carbon between two cor- 
rugated carbon blocks. 
The black patches in Fig. 
203 represent the carbon blocks. One of these blocks is sup- 
ported rigidly, and the other is attached to the diaphragm DD; 
and as the diaphragm moves inwards or outwards it produces 
increased or decreased compression of the granular carbon 
which produces a de- 
creased or increased elec- 
trical resistance. A bat- 
tery sends current through 
the granular carbon and 
through the primary coil 
of a small induction coil or 
transformer. The varying 
resistance of the granular 
carbon causes the battery current to rise and fall as the diaphragm 
moves inwards and outwards, and this rise and fall of battery 
current produces in the secondary coil of the transformer a cur- 
rent which flows in one direction and the other alternately as the 
transmitter diaphragm moves to and fro. 




Fig. 204. 
Telephone receiver (old style). 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 273 



The telephone receiver. — The simplest type of telephone re- 
ceiver is shown in Fig. 204. A coil of very fine insulated wire is 
wound around one end of a permanent steel magnet MM. The 
reversals of current from the distant transmitter flowing through 
this coil alternately strengthen and weaken the steel magnet, 



receiver 




line 



transmitter 




induction coil 
or transformer 




ground ground ■'^^ 

Fig. 205. 
Two-station telephone set (without call bells). 

and these variations of strength of the steel magnet cause the 
telephone diaphragm dd to move to and fro, thus reproducing 
the original sound. The most improved form of telephone 
receiver has a bipolar magnet. 

The simple telephone set. — Two telephone stations connec- 
ted up for talking are shown in Fig. 205. The ground return, 
as shown in Fig. 205, is replaced by wire return in Figs. 206 and 



receiver 



transmitter 

>.Hi|j|h-s, 




line 



hook 



line 




Fig. 206. 
Two-station telephone set — connections for ringing. 



19 



274 



ELECTRICITY AND MAGNETISM. 



207. To give a call at the distant station a small hand-operated 
dynamo is used to ring a bell, and the change from the connec- 
tions required to operate the bell to the connections required for 
the operation of the transmitter and receiver is made by the 
movement of the hook when the telephone receiver is taken from 
the hook. Figure 206 shows the hooks down, and the connec- 



recewer 



^=5 



line 



transmitter 



hook Y y 




line 




Fig. 207. 
Two-station telephone set — connections for talking. 

tions, as indicated by the full lines, are proper for operating the 
bell at either station. Figure 207 shows the hooks up, and the 
connections are proper for operating the transmitters and re- 
ceivers. 

195. Eddy currents. Lamination. — Figure 208 shows an end 
view of an iron rod surrounded by a wire ring. While the iron 

Hlamenf of 
iron 



Wire nng" 





Fig. 208. 
Current induced in wire ring. 



-iron rod 

Fig. 209. 

Current induced in filament of solid iron rod. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 275 

rod is being magnetized or demagnetized an electromotive force 
is induced in the ring, according to Art. 190, and an electric cur- 
rent is produced in the ring in the direction of the small arrows or 
in the opposite direction. Figure 209 shows the end of a larger 
iron rod. While the rod is being magnetized or demagnetized an 
electric current is produced in the circular filament of iron. The 
increasing or decreasing magnetism of the central portion C of the 
rod in Fig. 20Q has the same action on the filament of iron in Fig. 2og 
as the increasing or decreasing magnetism of the iron rod in Fig. 208 
has on the wire ring in Fig. 208. 

Every circular filament in an iron rod has more or less current 
induced in it while the rod is being magnetized or demagnetized. 
Thus the currents which are induced in a solid iron rod while it 
is being magnetized or demagnetized are in the directions of the 
arrows in Fig. 210 or in the opposite directions, and these currents 
are called eddy currents. One effect of these eddy currents is to 
make it impossible to magnetize or demagnetize a solid iron rod 
quickly and another effect is to generate heat in the rod. 






Fig. 210. 

Currents induced in 
solid iron rod. 



Fig. 211. 



Fig. 212. 



If the rod is a bundle of thin strips of sheet iron, as shown in 
Fig. 211, or a bundle of fine iron wires, as shown in Fig. 212, then 
the eddy currents (as shown in Fig. 210) cannot flow because the 
strips of iron in Fig. 211 and the iron wires in Fig. 212 are suffi- 
ciently insulated from each other by the thin coating of oxide 
which always covers the iron. 

Eddy currents are not only produced in a solid iron rod while 



276 



ELECTRICITY AND MAGNETISM. 



it is being magnetized or demagnetized, but eddy currents are 
also generally produced in a piece of solid iron (or in any solid 
piece of metal) which moves near a magnet. Thus if the cylinder 
A A , Fig. 141 , were solid and if it were set rotating as indicated by 
the curved arrows in Fig. 142, eddy currents would be produced 

solid iron cylinder^ 
rotating 




Fig. 213. 

Eddy currents in rotating solid cylinder; away from reader on right side, towards 

reader on left side. 

in it as indicated by the circles with dots and crosses in Fig. 213. 
If the cylinder is built up of thin sheet iron disks or stampings, 
these eddy currents cannot flow because the disks are sufficiently 
insulated from each other by films of iron oxide. 

An iron rod or core which is built up of stampings of thin 
sheet iron or of fine iron wires is said to be laminated. Armature 
cores of dynamos and transformer cores are always laminated. 

196. Inductance of a circuit. — Figure 214 shows a centrifugal 
pump (like fan blower) maintaining a current of water in a circuit 

pipe 




pump 



pipe 








wire 




iron core 


^ 


i 


p- battery 


P' 


" 


wire 





Fig. 214. 



Fig. 215. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 277 

of pipe. When the valve V is suddenly closed a momentary 
excessive force is exerted on the valve by the water. The moving 
water cannot be stopped instantly. To make this behavior of 
the water stream more clearly analogous to the behavior of the 
electrical current in Fig. 215 it is best to say that the moving water 
builds up momentarily an excessive pressure difference on the two 
sides' of the valve when the valve is suddenly closed 

Figure 215 shows a battery maintaining an electric current 
through a circuit. When the circuit is suddenly broken at p 
the stopping of the current builds up momentarily an excessive 
electrical pressure- difference or electromotive force across the 
break. The electric current cannot be stopped instantly. If 
the circuit contains a winding of wire on a laminated iron core, 
as shown in Fig. 215, the above described effect is so marked 
that the current continues to flow for a very short time across 
the break in the form of a visible spark. This effect is called 
the spark at break. 

It is the mass and velocity-value of the moving water that 
determine the violence of the "hammer" effect in Fig. 214; 
it is what is called the inductance of the circuit and the cur- 
rent-value that determine the violence of the spark at break in 
Fig. 215. 

The above described hydraulic analogue of the spark at break 
appeals to one largely because the picture of a circuit of pipe is 
like the picture of an electrical circuit; but one can more easily 
set up the equation of increasing or decreasing current by con- 
sidering the acceleration or deceleration of a moving body; and 
in the following discussion friction and electrical resistance are 
supposed to be non-existent, not that this supposition is necessary 
but that it avoids tedious and non-essential qualifications. 

A force F acting on a body An electromotive force E 

(which moves without friction) acting on a circuit (of which the 

causes the velocity v of the resistance is zero) causes the 

body to increase at a definite current i in the circuit to 



278 



ELECTRICITY AND MAGNETISM. 



rate 7— such that 
ai 

_ dv 

F = m — 

at 



1 r • ^^ 

increase at a definite rate — 



(83) 



such that 



E 



where m is the mass of the 
body. This is Newton's first 
law of motion as expressed by 
equation (i) of Art. 6. 

A moving body stores a cer- 
tain amount of energy which is 
given up when the body stops. 
This is the kinetic energy W 
of the body, and it is 



di 
Jt 



(84) 



W 
See Art. 46. 



_ 1 



mv^ 



(85) 



where L is what is called the 
inductance of the circuit. 



A circuit in which a current 
is flowing stores a certain 
amount of energy which is 
given up when the current 
stops. This is the kinetic 
energy W of the circuit and 
it is 

W = \Li^ (86) 



The inductance of a circuit is defined by equation (84), and 
a circuit has unit of inductance when one unit of electromotive 
force will cause the current in the circuit to increase at the rate 
of one unit per second. 

A circuit has an inductance of one henry when one volt will 
cause the 'current in the circuit to increase at the rate of one 
ampere per second. 

A circuit has an inductance of one abhenry when one abvolt 
will cause the current in the circuit to increase at the rate of one 
abampere per second. 

When L is expressed in henrys and i in amperes in equation 
(86), then W is expressed in joules. When L is expressed in 
abhenrys and i in abamperes, then, of course, W is expressed 
in ergs. 

dv 
* Let the reader remember that — is a single symbol meaning the rate of 

at 



change of v. 



MAGNETIC EFFECT OF THE ELECTRIC CURRENT. 279 

A very interesting experiment is to connect an ordinary iio- 
volt glow lamp in series with a large inductance as shown in Figs. 
216 and 217. If the inductance is made of fairly coarse copper 
wire its resistance will be negligible, and when connected to direct- 



direct current 
9upply main» 



lamp 


altemating-curreni ■ 

.0 


)laaip 


inductance 



nupply mahA 





inductamx 



Fig. 216. Fig. 217. 

current supply mains the lamp will be heated to normal bright- 
ness; the only effect of the inductance is to delay the heating of 
the lamp filament to a very slight extent when the circuit is first closed. 
If the lamp and inductance of Fig. 216 are connected to alter- 
nating-current supply mains, as shown in Fig. 217, the lamp 
filament is not perceptibly heated because an alternating electro- 
motive force does not produce a perceptible current through a 
large inductance. This effect of a large inductance in an alternat- 
ing-current circuit is called a choking effect ^ and an inductance is 
sometimes called a choke coil. 

A very considerable force is A very considerable electro- 
required to produce a percep- motive force is required to 
tible to and fro motion of a body produce a perceptible to and 
of large mass ; or a moderate to fro current (alternating cur- 
and fro force produces only an rent) in a circuit of large induc- 
impercep tible to and fro mo- tance ; or a moderate to and fro 
tion of a body of large mass, .electromotive force (an alter- 
especially if the reversals of the nating electromotive force) pro- 
force are rapid (of high fre- duces only an imperceptible 
quency). alternating current in a circuit 

of large inductance, especially 
if the frequency of the alter- 
nating electromotive force is 
high. 



280 ELECTRICITY AND MAGNETISM. 

197. Gas-engine ignition. — The older form of gas-engine 
igniter, the wipe-spark igniter, is an arrangement of which the 
essential features are shown in Fig. 215. An electric circuit 
containing a battery and an inductance* is broken at a point 
inside of the gas engine cylinder, and the spark at break ignites 
the mixture of gas and air (or the mixture of gasoline vapor and 
air) in the cylinder. 

The more usual form of gas engine igniter, the jump-spark 
igniter, is an induction coil, the primary circuit of which is closed 
at the instant that ignition is desired, and the vibration of the 
interrupter in the primary circuit of the induction coil produces 
a torrent of sparks across a short spark gap (inside of the gas 
engine cylinder) to which the secondary coil of the induction 
coil is connected. 

*The word inductance or inductance coil must not be confused with the term 
induction coil. 



CHAPTER XV. 



ELECTRIC CHARGE AND THE CONDENSER. 

198. The elimination of the water hammer effect by an air 
cushion. The elimination of the spark at break by a condenser. 

— The water hammer effect 
which is produced when a hydrant 
is suddenly closed is sometimes 
sufficiently intense to burst the 
pipe or injure the valve of the 
hydrant. In some cases, there- 
fore, it is desirable to protect 
the hydrant by an air cushion as ' 'i s^ _ 
indicated in Fig. 218. When " 
the hydrant in Fig. 218 is closed 
(however quickly) the moving water in the pipe is brought to 
rest gradually as it slowly compresses the air in the chamber CC. 

pipe 




hydrant 



-water 



Fig. 218. 




centrifugal pump 



check valve 



pipe 




Fig. 219. . ^ 

Figure 219 shows a centrifu- Figure 220 shows a battery 

gal pump maintaining a stream maintaining a "current of elec- 

of water through a circuit of tricity" through a circuit. If 

pipe. If the check valve is the circuit is broken at p, the 

suddenly closed, the water will electric current will continue 

continue for a short time to for a short time to flow through 

flow through the circuit into the circuit into the metal plate 

the chamber A and out of the A and out of the metal plate 

281 



282 



ELECTRICITY AND MAGNETISM. 



chamber B. This continued B. This continued flow of 

flow of the water into chamber the electric current into plate 

A and out of chamber B A and out of plate B causes 

causes a bending (a mechanical what we may think of as an 

stress) of the elastic diaphragm "electrical bending" (an elec- 



wire 




wire 



-~ battery 



wire 



DcZ 



wire 



B 



:;=iD 



Fig. 220. 

DD, and this bending soon trical stress) of the layer of 
stops the flow of water; then insulating material DD, and 



the diaphragm unbends and 
produces a reversed surge of 
water through the circuit of 
pipe. 



this "electrical bending" soon 
stops the flow of current; then 
the layer of insulating ma- 
terial "unbends" and produces 
a reversed surge of electric 
current through the circuit. 



The two metal plates A and B, Fig. 220, together with the 
layer of insulating material between them constitute what is 
called a condenser. A condenser is usually made of sheets of 



.1 f r f " 1 , 1 . '-^^r 



Fig. 221. 

tin foil separated by sheets of waxed paper. Thus the heavy 
horizontal lines in Fig. 221 represent sheets of tin foil, and the 
fine dots represent insulating material. In order that the follow- 
ing effects may actually be observed the condenser in Fig. 220 
must be made of a large number of sheets of tin foil and waxed 
paper. 



ELECTRIC CHARGE AND THE CONDENSER. 283 

The actual flow of current into the metal plate A and out 
of the metal plate B when the circuit is broken at p in Fig. 220 
is shown by the fact that no spark at break is produced when the 
condenser AB is connected, whereas a very perceptible spark 
at break is produced when the condenser AB is not connected. 

The reversed surge of current which takes place after the 
original current has been stopped in Fig. 220 may be shown as 
follows: Disconnect the condenser AB, make and break contact 
at p, hold a magnetic compass near one end of the core of the 
inductance* coil, and the core will be found to have retained a 
large portion of its magnetism; in other words, the cote will 
not have become by any means completely demagnetized when 
the circuit is broken and the current reduced to zero. Then 
connect the condenser AB, make and break the circuit at p 
as before, and again test the core of the inductance coil with a 
compass. The core will now be found to have lost nearly the 
whole of its magnetism because of the reversed surge of current. 

A reversed surge of current from the condenser in Figs. 201 
and 202 is the cause of the very quick demagnetization of the 
core of an induction coil. 

199. The momentary flow of current in an open circuit. Idea 
of electric charge. — ^When the metallic contact at p in Fig. 220 is 
broken the electric circuit remains closed as long as the current 
continues to flow acrOvSS the break in the form of a spark. The 
intensely heated air in the path of a spark is a conductor. When 
the condenser AB is connected as shown in Fig. 220, there is no 
spark at break, and the circuit is actually opened at the moment 
the contact at p is broken. The continued flow of current 
through the circuit after the contact at p in Fig. 220 is broken 
is an example of the momentary flow of current in an open circuit. 
The current continues for a very short time to flow through the 
open circuit into plate A and out of plate B after the circuit 
is broken at p, and the two plates A and B are said to become 
electrically charged. The plate into which the momentary current 

* A coil which has inductance. Do not confuse this term with induction coil. 



284 



ELECTRICITY AND MAGNETISM. 



flows is said to become positively charged, and the plate out of 
which the momentary current flows is said to become negatively 
charged. 

The flow of current in an open circuit may be shown by con- 
necting a small incandescent lamp and a condenser in series to 
alternating-current supply mains as shown in Fig. 223. With 
each reversal of the alternating supply- voltage a momentary 
current flows through the lamp, and the repeated pulses of current 



direct-current 
supply mains 



alternating-current 
supply mains 



1-0 



iamp 



Mi> 



lamp 



condenser 

Fig. 222. 



condenser 

Fig. 223. 



heat the lamp filament to incandescence. When the lamp and 
condenser are connected to direct-current supply mains, as shown 
in Fig. 222, a vsingle monentary current flows through the lamp 
when the connection is first made, but this single momentary 
pulse of current has no appreciable heating effect on the lamp 
filament. 

200. The function of choke coil and condenser in the lightning 
arrester. — Figure 225 shows the essential features of the lightning 
arrester for protecting a dynamo G from a lightning stroke. 
When the lightning strikes the trolley wire, a very large electro- 
motive force acts for a very short time between the trolley wire 
and ground, the spark gap breaks down, and the lightning dis- 
charge flows across the spark gap to earth. Meanwhile the large 
electromotive force has started an appreciable current through 
the choke coil, and nearly the whole of this current flows into one 
plate of the condenser and out of the other plate of the condenser 
to earth. But the current has been of such extremely short 



ELECTRIC CHARGE AND THE CONDENSER. 



285 



duration that the voltage across the condenser terminals (and 
across the dynamo terminals) has not risen to any considerable 



rubber cushion, 
hammer i 




\ wall 



trolley wire choke coil 



spark gap ^ 



rinnnnnr 



condenser 



ground 



Fig. 224. Fig. 225. 

Wall well protected from shock. Dynamo well protected from lightning stroke. 

value. Thus the dynamo A is protected from the action of a 
high voltage across its terminals. 

When the hammer strikes the When lightning strikes the 

ball in Fig. 224 it exerts a very trolley wire a very large voltage 

large force on the ball for a acts from trolley wire to ground 

very short time. for a very short time. 

This force sets the ball in This electromotive force sets 
motion. up a current through the choke 

coil. 

This motion continues for a This current continues for a 
relatively long time, slowly relatively long time, slowly 

"squeezing" the condenser,* 
that is, slowly charging the 
condenser. 

Therefore a comparatively 
small electromotive force is 
exerted across the condenser 
terminals and across the dy- 



squeezing the cushion. 



Therefore a comparatively 
small force is exerted against 
the wall for a comparatively 
long time. 

* Let one consider how unsuggestive it is to speak of charging a condenser and 
how extremely suggestive it is to speak of "squeezing" a condenser. See Arts. 
198 and 199. 



286 



ELECTRICITY AND MAGNETISM. 



The very large and short 
duration force which is exerted 
on the ball by the hammer is 
converted by the ball and 
cushion into a much smaller 
force of long duration as ex- 
erted against the wall. 



201. Electrostatic attraction. 



namo terminals for a com- 
paratively long time. 

The very large and short 
duration electromotive force 
from trolley wire to ground (due 
to the lightning) is converted 
by the choke coil and condenser 
into a much smaller electro- 
motive force of long duration 
as exerted across condenser and 
dynamo terminals. 

The electrostatic voltmeter. — 

When a momentary current flows into plate A and out of 
plate B in Fig. 220, the plates are said to become electrically 
charged, as stated in Art. 199, the plate into which the momentary 
current flows is said to become positively charged and the plate 
out of which the momentary current flows is said to become nega- 
tively charged. Two plates which have thus been oppositely 
charged attract each other when the intervening insulating material 
is a fluid like air or oil. 

This attraction between two oppositely 
charged metal plates is utilized in the elec- 
trostatic voltmeter which consists of a very 
delicately suspended metal plate and a sta- 
tionary metal plate, both carefully insulated. 
The electromotive force to be measured is 
connected to these plates, a momentary flow 
of current charges one plate positively and 
the other plate negatively, the suspended 
plate is moved by the electrostatic attraction 
between the plates, and a pointer attached 
to the movable plate plays over a divided 
scale. Figure 226 shows the essential features 
of an electrostatic voltmeter. The moving element of the instru- 




Westinghouse type 
electrostatic voltmeter. 



ELECTRIC CHARGE AND THE CONDENSER. 287 

ment consists of two very light metal vanes VV (seen edgewise 
in the figure) mounted on a pivot and carrying a pointer p, and 
the stationary element consists of two metal plates PP (also 
seen edgewise in the figure). The plates PP are connected 
together and to one terminal of the voltage to be measured, and 
the other terminal of the voltage is connected to the moving 
element VV by means of the controlling hair spring. 

Electrostatic attraction is familiar to every one. A hard- 
rubber comb, for example, is charged with electricity when it is 
passed through dry hair, at the same time the hair is oppositely 
charged, and each hair is attracted by the comb. 

202. Definition of the coulomb. — A current of water through 
a pipe is a transfer of water along the pipe. Let Q be the 
amount of water which during t seconds flows past a given point 
in the pipe, then the quotient Q/t is the rate of flow of water 
through the pipe, and this rate of flow may be spoken of as the 
strength / of the water current. If the strength I of the water 
current in cubic centimeters per second is given, then the amount 
of water flowing past a given point of the pipe in t seconds is 
given by the equation: 

Q = It 

in which / is the strength of the water current in cubic centi- 
meters per second, and Q is the number of cubic centimeters of 
water which flows past a given point of the pipe in t seconds. 

Similarly an electric current in a wire may be looked upon as 
the transfer of "electricity" along the wire, and the quantity 
Q of "electricity" which flows past a point on the wire during t 
seconds may be defined as the product of the strength of the 
current and the time, that is we may write: 

Q = It (87) 

in which I is the strength of the current in amperes, and Q 
is the quantity of electricity which flows past a point on the wire 
during / seconds. It is evident from equation (87) that the 



288 ELECTRICITY AND MAGNETISM. 

product of amperes and seconds gives quantity of electricity, 
and therefore the unit of quantity of electricity is most conve- 
niently taken as one ampere-second, meaning the amount of elec- 
tricity which during one second flows past a point on a wire in 
which a current of one ampere is flowing. The ampere-second is 
usually called the coulomb. One ampere-hour is the quantity of 
electricity carried in one hour by a current of one ampere. 

The c.g.s. unit of charge in the "electromagnetic system" is 
the amount of charge carried in one second by a current of one 
abampere, and it is called the abcoulomh. 

The c.g.s. unit of charge in the "electrostatic system" is that amount of charge 
which if concentrated on a very small body would repel a similar charge with a 
force of one dyne at a distance of one centimeter. The "electrostatic system" 
system of units is not used in this text. 

203. Measurement of electric charge. The ballistic galvanom- 
eter. — A very large quantity of electric charge may be determined 
by observing the time during which the charge will maintain a 
sensibly constant measured current. Thus, a given storage cell 
can maintain a current, say, of ten amperes for eight hours so 
that the discharge capacity of the storage battery is equal to 
eighty ampere-hours. The quantities of charge which are most 
frequently encountered in the momentary flow of electric current 
in open circuits are, however, exceedingly small. For example, 
the terminals of a given condenser are connected to no- volt 
direct-current supply mains, and the momentary flow of current 
represents the transfer of, say, 0.000 1 of a coulomb which corre- 
sponds to a flow of one ampere for a ten-thousandth of a second. 
It is evident that such a small amount of electric charge cannot be 
measured by the method above suggested. Such small quanti- 
ties of electric charge are measured by means of the ballistic 
galvanometer. This galvanometer is an ordinary D'Arsonval 
galvanometer such as described in Art. 141. When a momentary 
pulse of current is sent through such a galvanometer, the coil is 
set in motion, and a certain maximum deflection or throw of the 
coil is produced. Let d be the measure of this momentary 
maximum deflection or throw on the galvanometer scale. A 



ELECTRIC CHARGE AND THE CONDENSER. 289 

certain amount of charge Q is represented by the momentary 
pulse of current, and this amount of charge is proportional to the 
throw d. That is, we may write: 

Q = kd (88) 

in which ^ is a constant for the given galvanometer, and it is 
called the reduction factor of the galvanometer. 

The value of the reduction factor k is generally determined in 
practice by sending through the galvanometer a known amount 
of charge Q and observing the throw d produced thereby. 

204. The capacity of a condenser. — ^A ballistic galvanometer 
BG, a condenser and a number of dry cells are connected as 
shown in Fig. 227. One terminal of the condenser is connected 




glass handle 

Fig. 227. 

to a flexible wire which is fixed to the end of a glass handle. By 

touching the wire W to the point b, the electromotive force E 

of one dry cell acts upon the condenser, and the momentary flow 

of current which charges the condenser produces a throw of the 

ballistic galvanometer. The condenser can then be discharged 

by touching the wire W to the point a. By touching the wire 

W to the point c, the electromotive force 2E of two dry cells 

acts upon the condenser, and the momentary flow of current 

which charges the condenser produces a throw of the ballistic 

galvanometer. The condenser may then be discharged as before. 

By touching the wire W to the point d, the electromotive force 

3^ of three dry cells acts upon the condenser, and the momentary 
20 



290 ELECTRICITY AND MAGNETISM. 

flow of current which charges the condenser causes a throw of 
the balUstic galvanometer; and so on. In this way the throws 
of the ballistic galvanometer may be observed when the con- 
denser is charged by an increasing series of voltages E, 2E, 3£, 
4E, and so forth, and it is found that the throw of the ballistic 
galvanometer becomes larger and larger in proportion to the 
voltage. But the throw of the ballistic galvanometer is pro- 
portional to the charge which is drawn out of one plate and forced 
into the other plate of the condenser. Therefore the amount of 
charge which is drawn out of one plate and forced into the other 
plate of a condenser is proportional to the electromotive force which 
acts upon the condenser. Therefore we may write: 

Q=CE (89) 

where Q is the quantity of charge which is drawn out of one 
plate and forced into the other plate of a condenser when an 
electromotive force of E volts is connected so as to act upon the 
condenser, and C is a constant for a given condenser. The 
factor C is adopted as a measure of what is called the capacity 
of the condenser. 

It is evident from the above equation that C, the capacity 
of a condenser, is expressed in coulombs-per-volt. One coulomb- 
per-volt is called a farad, that is to say a condenser has a capacity 
of one farad when an electromotive force of one volt will draw one 
coulomb out of one plate of the condenser and force one coulomb 
into the other plate of the condenser. 

Condenser capacities as usually encountered in practice are 
very small fractions of a farad. Thus the capacity of a condenser 
made by coating with tin foil the inside and outside of an ordi- 
nary one-gallon glass jar would be about one five-hundred- 
millionth of a farad, or 0.002 of a microfarad. A microfarad is a 
millionth of a farad, and in practice capacities of condensers are 
usually expressed In microfarads. 

The approximate dimensions of a one-microfarad condenser are 
as follows: 501 sheets of tin foil separated by sheets of paraffined 



ELECTRIC CHARGE AND THE CONDENSER. 291 

paper 0.05 centimeter in thickness, the overlapping portions of 

the tin foil sheets being 25 . 

centimeters X 25 centimeters, / ^1 

as shown in Fig. 228. / ^| 

Two pieces of metal of any shape I :«/ 

separated by insulating material I ^; 

constitute a condenser; the only j ^! 

reason for using sheets of metal j 1 ^ 

with thin layers of insulating ma- ^ j 

.77. • . 7 . • 7 I 25 centimeters I 

tertal between is to obtain a large ^ -^** 

. •, Fig. 228. 

capacity. 
Example showing the use of the ballistic galvanometer. — 

A condenser of which the capacity is C farads is charged by an 
electromotive force of E volts, and discharged through a 
ballistic galvanometer ; and the observed throw of the galvanom- 
eter is d scale divisions. Then: 

CE = kd (i) 

This equation is evident when we consider that CE is the charge 
which has been drawn out of one plate of the condenser and 
forced into the other plate by the charging electromotive force 
E, and this amount of charge flows through the galvanometer 
when the condenser is discharged ; furthermore the charge which 
flows through the ballistic galvanometer is equal to kd according 
to Art. 203. 

Another condenser of which the capacity is C farads is 
charged by the same electromotive force E, and discharged 
through the ballistic galvanometer; and the observed throw of 
the galvanometer is d' scale divisions. Then : 

CE = kd' (ii) 

Dividing equation (i by equation (ii) member by member, we 

get 

C d d 

from which C can be calculated if C is known. 



292 



ELECTRICITY AND MAGNETISM. 



The standard condenser. — A condenser of which the capacity 
has been carefully measured* is called a standard condenser. 
Thus, if C in equation (iii) is the known capacity of a standard 
condenser, the value of C may be calculated. 

A standard condenser may be used to determine the reduction 
factor of a ballistic galvanometer. Thus, if C is the known 
capacity of o standard condenser, and if E is a known voltage, 
then everything but k is known in equation (ii), so that the 
value of k may be calculated. 

205. Inductivity of a dielectric. — The insulating material be- 
tween the plates of a condenser is called a dielectric ^ Indeed, 
the insulating material between any two oppositely charged 
bodies is called a dielectric. The capacity of a condenser depends 
upon the size of the plates, upon the thickness of the dielectric 
and upon the nature of the dielectric. The dependence of the 
capacity of the condenser upon the nature of the dielectric is a 
matter which must be determined purely by experiment. Thus 



fcrire 



air 



air 

Fig. 229. 



Wire 



wire 



B 




wire 



Fig. 230. 



Fig. 229 represents two metal plates with air between them, and 
Fig. 230 represents the vsame plates immersed in oil. The dis- 
tance between the plates is understood to be the same in Figs. 229 
and 230. Let C be the capacity of the condenser in Fig. 229 
with air as the dielectric, and let C be the capacity of the con- 
denser in Fig. 230 with a given kind of oil as the dielectric. The 

* Methods of measuring capacity are described in Gray's Absolute Measure- 
ments in Electricity and Magnetism, Vol. I, pages 418-450. 



ELECTRIC CHARGE AND THE CONDENSER. 293 

ratio C'jC is called the inductivity* of the oil. Thus the induc- 
tivity of kerosene is about 2.04, that is, the capacity of a given 
condenser is 2.04 times as great with kerosene between the plates 
as with air between the plates. The accompanying table gives 
the inductivities of a few dielectrics. 

TABLE. 
Inductivities of Various Substances. 

Crown glass (according to composition) 3.2 to 6.9 

Flint glass (according to composition) 6.6 to 99 

Hard rubber 2.08 to 3.01 

Sulphur (amorphous) 3.04 to 3.84 

Paraffin 2.00 to 2.32 

Shellac 2.74 to 3.67 

Ordinary rosin 2.48 to 3.67 

Mica (according to composition) 5.66 to 10 

:-2troleum about 2.04 

Water about 90. 

206. Dependence of capacity of a condenser upon size and 
distance apart of plates. — When the dielectric of a condenser is 
of uniform thickness and when the metal plates are large as 
compared with their distance apart (thickness of dielectric), 
then the capacity C of the condenser is proportionalf to a/x 
for a given dielectric, where x is the thickness of the dielectric 
and a is the area of the sheet of dielectric between the plates. 
Therefore, if we choose a given dielectric, we may write 

C = B-- (i) 

X 

In which B is a constant. When x is expressed In centimeters, 
and a in square centimeters; when air Is chosen as the dielec- 
tric; and when C Is expressed in farads; then the value of B 

* What is here called the inductivity of a dielectric is sometimes called dielectric 
constant, or specific capacity of a dielectric, or specific inductive capacity of a dielectric. 

t It can be shown from almost purely geometrical considerations that C is 
proportional to a]x, but it is sufficient to accept this proportional relation as the 
result of experiment. The value of the proportionality factor B must be deter- 
mined by experiment directly or indirectly. 



294 ELECTRICITY AND MAGNETISM. 

as found by experiment is 884 X io~^^. Therefore we have: 

Cin farads = 884 X lO'^^ X ^ (gOa) 

X 

When a dielectric whose inductivity is k is used instead of air, 
the capacity of the condenser is k times as great, or: 

ka 

C'ln farads = 884 X IQ-^' X — (gob) 

in which C is the capacity in farads of a condenser of which 
the plates are separated by a layer of dielectric x centimeters 
thick and a square centimeters in area (between the plates), 
and k is the inductivity of the dielectric. The meaning of a 
may be understood with the help of Fig. 228. If there are 501 
sheets of tin foil there will be 500 sheets of dielectric, and a will 
be equal to 500 X 25 centimeters X 25 centimeters, which is 
312,500 square centimeters. 

207. The work done by an electromotive force E in pushing 
a given amount of charge, Qy through a circuit. — When Q 
coulombs of electric charge flow through a battery of which 
the electromotive force is E, the amount of work W done by 
the battery is EQ joules. That is: 

W = EQ (91) 

This is evident from the following considerations. Imagine a 
current / flowing through the battery; then EI watts is the 
rate at which the battery does work, and Elt joules is the 
amount of work done in / seconds. But the product // is the 
amount of charge Q (in coulombs) which has been pushed 
through the circuit. Therefore the work done, namely Elt 
joules, is expressible as EQ joules. 

208. The potential energy of a charged condenser. — A charged 
condenser represents a store of potential energy in much the 
same way that a stretched spring or the distorted diaphragm DD 
in Fig. 219 represents a store of potential energy, and before con- 



ELECTRIC CHARGE AND THE CONDENSER. 



295 



sidering the amount of potential energy in a charged condenser 
it is helpful to consider the amount of potential energy in a 
stretched spring. 

Let g represent the elongation of a spring due to a stretching 
force e as shown in Fig. 231. As is well known g is proportional 




stretched spring 

Fig. 231. 

to e; therefore if we plot corresponding values of g and e as 
abscissas and ordinates respectively, we will get a straight line 
cc as shown in Fig. 232. 

Consider the total amount of work W which is done while the 
spring is being stretched from q = o to q = Q, and while the 
stretching force is increasing from e = o to e = E. The aver- 
age value of the stretching force is J£, as may be understood 
from Fig. 232, and the work done is equal to the product of the 
total stretch Q and the average stretching force J£. That is: 

W=iEQ 

and this work W is stored in the stretched spring as potential 
energy. Thus a stretch of 3 centimeters {= Q) is produced in 
a large spring while the stretch- 
ing force rises from zero to 
2,000,000 dynes (= E). The 
average value of the stretching 
force is 1,000,000 dynes (= |£), 
and the work done is 3,000,000 
ergs (= iEQ). This work is 
stored in the stretched spring as 
potential energy. 



axis of e 




296 ELECTRICITY AND MAGNETISM. 

Similarly, a condenser is charged by applying it to an electro- 
motive force which begins at zero and rises to E volts, and the 
amount of work W which is done in charging the condenser is 
equal to ^EQ, where J£ is the average value of the charging 
electromotive force, and Q is the total charge which is drawn 
out of one plate of the condenser and pushed into the other 
plate. This statement is in accordance with equation (91) of 
Art. 207. Therefore; 

W = \EQ (92) 

where W is the potential energy of a charged condenser, E is 
the voltage acting on the charged condenser, and Q is the charge 
which has been drawn out of one plate of the condenser and 
pushed into the other plate; W is expressed in joules when E 
is in volts and Q in coulombs. 

We may substitute CE for Q in equation (92), according to 
equation (89) of Art. 204, and we get: 

W=hCE? . (93) 

or we may substitute QjC for E in equation (92), according to 
equation (89) of Art. 204, and we get: 

TF^J^ (94) 

Following is a rigorous derivation of equation (94) as applied to a stretched 
spring. Let q be the elongation of the spring when the stretching force is e. 
Then g and e are proportional, so that: 

q = Ce (i) 

where C is a constant for the given spring. Let A5 be the added elongation 
due to an increment Ae of the stretching force, and let APT be the work done on 
the spring to produce the added elongation. Then: 

APT is greater than g'Ag 

and 

APF is less than (e + Ae)-A5 
or 

APF 

is greater than e and less than {e + Ae). 

Ag 

Therefore APT/A^ approaches g as a limit when Ae and A5 both approach zero 
or, using differential notation, we have dW/dq = e; or, using the value of e from 



ELECTRIC CHARGE AND THE CONDENSER. 297 

equation (i), we have: 

Now the potential energy W of the spring when its elongation is Q, is the 
amount of work done in stretching the spring from q = o to q — Q, and this is 
found by integrating equation (ii) from q = o to q = Q, which gives: 

1 02 

W =-^ (iii) 

2 C 

209. Disruptive discharge. Dielectric strength. — When the 
electromotive force which charges a condenser is increased more 
and more, the dielectric of the condenser is eventually broken 
down; this break-down occurs in the form of an electric spark, 
it discharges the condenser, and it is called a disruptive discharge. 
By a condenser is here meant two metal bodies of any shape 
separated by insulating material. The electromotive force re- 
quired to break down a dielectric depends upon three things, 
namely, {a) the shape of the metal bodies, (b) the minimum dis- 
tance* between the metal bodies, and (c) the nature of the dielec- 
tric. The dependence upon the shape of the metal bodies is 
illustrated by the fact that a given electromotive force will 
produce a much longer spark between points than between flat 
metal surfaces. In the whole of the following discussion the 
dielectric is assumed to be between flat metal plates. 

When the dielectric is perfectly homogeneous like air or oil, 
the voltage required to break it down is very nearly proportional 
to its thickness, and the voltage required to break down such a 
dielectric divided by the thickness of the dielectric is called the 
specific strength of the dielectric. Thus the specific dielectric 
strength of air is about 35,000 volts per centimeter. When the 
dielectric is non-homogeneous the voltage required to break it 
down is not even approximately proportional to its thickness. 
The most familiar example of a non-homogeneous dielectric is 
the material which is used for insulating the windings of dynamos 

* This is not true when the distance is very small or when the bodies are in a 
very good vacuum. 



298 ELECTRICITY AND MAGNETISM. 

and transformers. This material is made up of layers of cloth 
and varnish and mica with occasional layers of air. 

If a tank is made with one wall of porous material like unglazed 
earthenware, the pressure of the fluid in the tank has three 
important effects upon the wall, namely, (a) fluid soaks through 
the wall at a certain rate, {b) the wall is slightly elastic and it 
yields a little to the fluid pressure, and (c) the wall has a certain 
ultimate strength and it will burst if the pressure exceeds a 
certain amount. Similarly the electromotive force which acts on 
a condenser has three important effects upon the dielectric of 
the condenser, naniely, (a) a certain amount of electric current 
"soaks" through the dielectric as it were, because the dielectric 
is an electrical conductor although a very poor one, (&} the 
dielectric has a certain amount of electrical "elasticity" (induc- 
tivity as it is properly called), and it "yields" a little to the 
electromotive force and allows a certain amount of charge to be 
drawn out of one plate and forced into the other plate of the con- 
denser, and (c) the dielectric has a certain ultimate strength and 
it will be ruptured if the electromotive force exceeds a certain 
amount. 

210. The spark-gauge. — The electromotive force required to 
produce a spark between polished metal spheres in air depends 
upon the length of the air gap between the spheres and upon the 
diameter of the spheres; and the accompanying table gives the 
observed sparking voltages corresponding to different lengths of 
air gap and different diameters of spheres. 

The spark-gauge is an arrangement for measuring an electro- 
motive force by observing the length of spark it will produce. 
As an example consider the following test of the break-down 
voltage of the rubber insulation on a wire. The arrangement 
is shown in Fig. 233. The two spheres BB of the spark-gauge 
are connected to a high- voltage electric machine,* one of the 
spheres is connected to the metal core of the wire to be tested, 
and a wire from the other sphere is wrapped around the outside 

* See Art. 223. 



ELECTRIC CHARGE AND THE CONDENSER. 



299 






to high voltage 
electric tnachine 



B 



of the insulation to be tested, as shown. The spheres BB are 
near together at the start, and they are slowly separated until 
the spark breaks through the in- 
sulation on the wire and ceases to 
jump between the spheres. For 
example the air gap between the 
spheres was increased to 0.6 centi- 
meter before the insulation on the 
wire broke down, and the spheres 
were each 2 centimeters in diam- 
eter; therefore the break-down 
voltage, as taken from the table, 
was 20,400 volts. Break -down tests 
are nearly always made in practice 
by using alternating voltage from a 

step-up transformer, and the spark-gauge usually has needle 
points instead of polished metal spheres. 




Fig. 233. 
Spark-gauge diagram. 



TABLE* 
Sparking Voltages in Air at i8° C. and 745 MM. Pressure. 





Between polished 


Between polished 


Between polished 


Between polished 


Spark gap in 


metal spheres 0.5 


metal spheres i.o 


metal spheres 2 


metal spheres 5 


centimeters- 


centimeter diameter. 


centimeter diameter 


centimeters diameter. 


centimetersdiameter. 




Volts. 


Volts. 


Volts. 


Volts 


O.I 


4.830 


4,800 


4,710 




0.2 


8.370 


8,370 


8,100 




0.3 


11.370 


11.370 


11.370 




0.4 


13,800 


14,400 


14,400 




0.5 


15,600 


17,400 


17.400 


18,300 


0.6 


17,100 


19,800 


20,400 


21,600 


0.7 


18,300 


21,900 


23,100 


24,600 


0.8 


18,900 


24,000 


26,100 


27,300 


0.9 


19,500 


25,500 


28,800 


30,000 


I.O 


20,100 


27,000 


31,200 


32,700 


I.I 


20,700 




33.300 


35.700 


1.2 


21,000 




35.400 


38,400 


1-3 


21,600 




37,200 


41,100 


1.4 


21,900 




38,700 


43.800 


1-5 


22,200 





40,200 


46,200 


1.6 






41,400 


48,600 



* A table giving the sparking voltage between needle points is given in Franklin 
& Esty's Elements of Electrical Engineering, Vol. II, page 44, The Macmillan Co., 
1908. 



300 



ELECTRICITY AND MAGNETISM. 



211. Electric oscillations. — ^When a condenser is charged by 
suddenly connecting it to a battery, or when a charged condenser 
is discharged by connecting its terminals with a wire, the current 
surges back and forth through the circuit. These back and forth 
surges of current are called electrical oscillations. This matter 
can be most clearly illustrated by showing the analogy between 
electrical oscillations and mechanical oscillations as follows : 



initial 
position 



equilibrium 
position 



extreme 
position 





unstretcked 
spring 



I 

j 

la 



-,±. 



\2Q 



Fig. 234. 



wire 



^ 
«> 



•o 



condenser 



C 

Fig. 235. 



V 



Figure 234 represents an un- Figure 235 represents an un- 

stretched spring, the attached charged condenser, the battery 

weight being supported, let us circuit being broken at p. 
say, by the hand. 



ELECTRIC CHARGE AND THE CONDENSER. 



301 



If the hand is removed, the 
full pull of gravity E will be- 
gin at once to act upon the 
weight L, and the velocity / 
of the weight will continue to 
increase so long as the down- 
ward pull of gravity is greater 
than the reacting pull due to 
the increasing stretch of the 
spring. 

When the spring is stretched 
by a certain amount Q, the re- 
acting pull of the spring is 
equal to the pull of gravity, 
but at this instant the velocity 
I of the weight has reached a 
maximum value. 

Consequently the weight 
goes on moving downwards, 
but the reacting pull of the 
stretched spring now exceeds 
the downward pull of gravity. 

Therefore the velocity I of 
the weight begins to decrease. 

When the velocity of the 
weight has been thus reduced 
to zero, the stretch of the 
spring has reached a certain 
value 2(2 (if there has been 
no friction loss of energy). 

The movement of the weight in Fig. 
234 is assumed to be frictionless for the 
sake of simplicity of statement. 



If the circuit is closed, the full 
electromotive force E of the 
battery will begin at once to 
act upon the circuit, and the 
current / in the circuit will 
continue to increase so long as 
the electromotive force of the 
battery is greater than the re- 
acting electromotive force due 
to the increasing charge on 
the condenser. 

When the condenser is 
charged to a certain extent Q, 
the reacting electromotive force 
of the condenser is equal to the 
electromotive force of the bat- 
tery, but at this instant the 
current I in the circuit has 
reached a maximum value. 

Consequently the current 
continues to flow, but the re- 
acting, electromotive force of 
the charged condenser now 
exceeds the electromotive force 
of the battery. 

Therefore the current I in 
the circuit begins to decrease. 

When the current has been 
thus reduced to zero, the 
charge of the condenser (posi- 
tive charge on one plate, nega- 
tive charge on the other) has 
reached a certain value 2Q 

The resistance of the connecting 
wires in Fig. 235 is assumed to be zero 
for the sake of simplicity of statement. 



302 



ELECTRICITY AND MAGNETISM. 



Then the reacting pull of the 
stretched spring starts the 
weight moving upwards. 



(if there has been no resistance 
loss of energy). 

Then the reacting electro- 
motive force of the charged 
condenser starts the current 
flowing in a reverse direction 
through the circuit. 
The weight therefore moves The current therefore surges 
repeatedly up and down until it repeatedly back and forth 
finally comes to rest with the through the circuit until it 
spring stretched so as to give a finally dies away with the con- 
reacting pull equal to the denser charged so as to give a 
downward pull of gravity. reacting electromotive force 

equal to the electromotive 
force of the battery. 

212. The disruptive-discharge as a means for exciting electric 
oscillations. — The method described in connection with Fig. 235 
for producing electric oscillations is never used in practice; a 
more satisfactory method is as follows: A condenser C, Fig. 236 








to 

high voltage 
supply 




C ^0 




b 


spark gap 




•c* 



Fig. 236. 

(usually consisting of a number of glass jars with coatings of tin 
foil inside and outside, called Leyden jars), is connected to the 
high-voltage terminals a and 6 of a step-up transformer. As 
the voltage* between a and b rises the condenser becomes 
charged, eventually the spark gap G breaks down, and then the 
charge on the condenser surges hack and forth through the inductance 
coil L and across the gap G until the energy of the charge is dis' 

* Alternating voltage, of course. 



m 



ELECTRIC CHARGE AND THE CONDENSER. 303 

sipated. When the back and forth surges cease, the air gap G 
quickly cools* and regains its insulating power, the condenser is 
again charged until the gap G breaks down, and another series 
of surges takes place, and so on. 

The Hertz oscillator. — Two brass rods A and B, Fig. 237, 
have a spark gap G between them. The rods are connected to 
the high- voltage terminals of an induction coil, and at each 
interruption of the primary circuit of the induction coil the rods 
A and B are charged sufficiently to produce a spark across the 
air gap G. This spark is a sudden break- 
down of the insulation of the gap, and this 
break-down is followed by a back and forth ^a__ 

surging of current across the gap and along f*> J^^inals of 

high voltage , 
the rods A and B. induction coil \ 

A condenser C and an inductance coil L 5 " 

arranged as shown in Fig. 236 constitute what 
is called an electric oscillator. Also the ar- p^g 237, 

rangement of the two rods A and B in 
Fig. 237 is an electric oscillator. The type of oscillator shown 
in Fig. 237 was devised by Heinrich Hertz in 1888, and this 
type of oscillator gives off a large portion of its energy in the 
form of electric waves. 

The use of the electric oscillator in wireless telegraphy. — Figure 
238a shows the essential features of a simple type sending 
station for wireless telegraphy, and Fig. 238^ shows the essential 
features of a receiving station. f The arrangement abCLG in 
Fig. 238a is the electric oscillator which is shown in Fig. 236. 
The two coils L and S constitute a transformer. The back and 
forth surging of the oscillating current in coil L induces a high- 
frequency electromotive force in the coil 5, this electromotive 
force causes high-frequency current to surge up and down in the 
antenna, and electric waves pass out from the antenna. These 

* The air in the path of an electric spark owes its electrical conductivity not 
only to high temperature, but also, and indeed chiefly, to the fact that the air 
molecules are broken up into charged atoms which are called ions. 

t The arrangement for "tuning" is not shown in Fig. 238&. 



304 



ELECTRICITY AND MAGNETISM. 



electric waves produce up and down surges of current (very weak) 
in the antenna at the receiving station, these up and down surges 
of current flow through the primary coil 5 of a transformer in 



<mtena 



antenna 




ground 




telephone 



ground 



Fig. 238a. 



Fig. 238&. 



Fig. 2385, and every alternate half -wave of the secondary current 
i flows through the crystal rectifier CM and through a telephone 
receiver. 

The antenna as usually constructed consists of a horizontal 
band of parallel wires stretched between two cross-pieces and 
supported by two masts, as shovvn in Fig. 238c. The crystal 
rectifier consists of a fragment of a crystal of galena C resting 
lightly on a metal plate M* 



•insulator 



mast 



^^insulatop 



band of wires 



Fig. 238c. 



mast 



213. The electric field. — ^When a momentary electric current 
flows through an open circuit, certain important effects are pro- 

* A good treatise is Wireless Telegraphy by Zenneck, translated by A. E. Seelig; 
McGraw-Hill Book Co., New York, 1915. 



ELECTRIC CHARGE AND THE CONDENSER. 305 

duced in the gap which breaks the circuit. In order that these 
effects may be easily observed a very high voltage must be used. 
The most convenient device for generating a high voltage is the 
influence electric machine which is described in Art. 223. Figure 
239 shows two brass balls which have been charged by a momen- 




Fig. 239. 

tary electric current drawn out of one ball and pushed into the 
other by an influence machine. When an ordinary wooden tooth- 
pick suspended by a fine thread is placed in the region between 
the balls, the tooth-pick points in a definite direction at each 
point very much as a magnetic needle points in a definite direc- 
tion at each point when it is placed between magnet poles. The 
short black lines in Figs. 239 and 240 represent the various 
positions of the tooth-pick. 

The behavior of the tooth-pick shows that the whole region sur- 
rounding the charged metal balls in Fig. 239 is in a peculiar 
condition, and this region is called an electric field. The direction 
of the electric field at each point is indicated by the direction 
of the tooth-pick when it is placed at that point, and lines 
drawn through the electric field so as to be, at each point, in 
the direction of the field at that point are called the lines of 
21 



3o6 



ELECTRICITY AND MAGNETISM. 



force of the electric field. Figure 240 shows the lines of force 
of the electric field between two charged flat metal plates. The 
lines of force in the region between the plates are straight lines, 
and the electric field is said to be uniform. 





Fig. 240. 

214. Intensity of electric field. — It would be permissible to 
adopt arbitrarily the ratio Ejx as a measure of the intensity of 
the uniform electric field between the flat metal plates in Fig. 240, 
E being the electromotive force between the plates and x 
being the distance between the plates. Thus the intensity of the 
electric field would be expressed in volts per centimeter or volts 
per inch. It is desirable, however, to base the definition of elec- 
tric field intensity upon some observable effect as in the 
following discussion. 

Two metal plates A and B, Fig. 241, are connected to an 
electric machine giving a high electromotive force E. The 
electric machine is represented in Fig. 241 as a battery for the 
sake of clearness. A small metal ball h is suspended between 
A and 5 by a silk thread. If this ball is started it continues 
to vibrate back and forth from plate to plate. 

Regarding the behavior of the vibrating ball the following 
statements may be made: 

(a) Work evidently is done to keep the ball h oscillating 
back and forth, and this work is evidently done by the battery. 



ELECTRIC CHARGE AND THE CONDENSER. 



307 



tipirc 



silk thread 







{b) The only way the battery can do work is by continuing 
to draw charge out of one plate and push it into the other plate. 
It is evident therefore that the ball carries charge back and forth 
between the plates. 

(c) The successive movements of the ball are similar, and there- 
fore if the ball carries charge 

at all it must carry a defi- 
nite amount each time it 
moves across. Let this defi- 
nite amount of charge be 
represented by q ; this charge 
is positive when the ball 
moves from A to B, and 
negative when it moves from 
B to A. At each move- 
ment of the ball the battery 
supplies the amount of charge 
q, drawing it out of plate B 
and pushing it into plate A . 
Therefore at each movement 

of the ball the battery does an amount of work Eq according 
to equation (91) of Art. 207. 

(d) Let F be the average mechanical force acting on the ball 
b while it is being pulled across from plate to plate. Assuming 
the ball b to be very small in diameter, it moves the distance x 
in traveling from plate to plate. Then Fx is the amount of 
work done on the ball while it moves from plate to plate. 

(e) The work Eq done by the battery during one movement of 
the ball is equal to the mechanical work Fx done on the ball, 
therefore we have Fx = Eq, or 




battery 

Fig. 241. 
The ball b oscillates to and fro. 



^ E 

F = —' q 

X 



(i) 



Any region in which a charged body is acted upon by a force* 

* A force which depends upon the charge on the body and which does not exist 
when the body is not charged. 



308 ELECTRICITY AND MAGNETISM. 

is called an electric field. Thus the region between A and B 
in Fig. 241 is an electric field because the charged ball b is acted 
upon by the force F. 

The force F with which an electric field pulls on a charged 
body placed at a given point in the field is proportional to the 
charge q on the body so that we may write : 

F=h (95) 

in which / is the proportionality factor, and it is called the 
intensity of the electric field at the point. 

From equations (i) and (95) it is evident that the intensity 
of the electric field between the plates A and B in Fig. 241 is: 

/ = f (96) 

that is, the intensity of the electric field between the plates is 
equal to the electromotive force between the plates divided by 
the distance between the plates. In the above discussion F is 
spoken of as the average force acting on the ball h in Fig. 241 
while the ball is moving from plate A to plate B. As a matter 
of fact this force is constant if the ball h is very small. 

Direction of electric field at a point. — The direction of an 
electric field at a point is the direction in which the field would 
pull on a positively charged body placed at that point. 

Tension of the lines of force in an electric field. — Two op- 
positely charged metal plates attract each other as stated in Art. 
201. Thus the oppositely charged plates in Fig. 240 attract 
each other. This attraction may be thought of as due to a 
tension of the lines of force; that is, the lines of force may be 
thought of as if they were filaments of rubber stretching from 
plate to plate and pulling the plates towards each other. 

If the lines of force in an electric field are like stretched fila- 
ments of rubber one would expect the lines of force to pull 
outwards on every part of the surface of a charged body. In 
fact each part of the surface of a charged body is pulled out- 



ELECTRIC CHARGE AND THE CONDENSER. 



309 




Fig. 242. 



wards by the surrounding electric field. This outward pull 
may be beautifully shown by pouring melted rosin in a thin 
stream from a metal ladle which is supported by an insulated 
handle and connected to one terminal of an electric machine. 
The lines of force which emanate 
from the lip of the metal ladle 
pull the melted rosin into ex- 
tremely fine jets which shoot 
straight outwards from the lip. 
These jets congeal in the form 
of excessively fine fibers which 
float about in the air. 

215. The idea of electric charge 
as the ending of electric lines of 
force. — Figure 242 represents two 
metal bodies A and B to which 
a battery is connected as shown. 
The battery draws a certain amount of charge out of one body 
B and forces it into the other body A , and the entire surround- 
ing region becomes an electric field, the lines of force of which 
are shown in the figure. The positive charge on body A may he 
thought of as the beginning of the lines of force, and the negative 
charge on body B may be thought of as the ending of the lines of 
force, the directions of the lines of force being indicated by the 
arrow heads in the figure. 

216. The pith-ball electroscope. — The presence of an electrical 
field may be shown by the behavior of a suspended wooden tooth- 
pick as described in Art. 213, and such a device may therefore 
be called an electroscope. A more sensitive electroscope is made 
by suspending a small pith ball by a very fine slightly conducting 
thread. When a charged body is brought near to such a sus- 
pended pith ball the ball becomes charged as indicated by the 
lines of force in Fig. 243 (see Art. 215), and the lines of force 
from the charged body to the ball pull the ball towards the body 



310 



ELECTRICITY AND MAGNETISM. 



as shown. The suspending thread may be made sHghtly con- 
ducting by soaking it in a dilute sal t solution and al lowing it to dry. 




pith ball 



Fig. 243. 

217. Electric charge resides wholly on the surface of a metal 
body. — Experiment shows that to whatever degree a hollow metal 
shell may be charged, no effect of the charge can be observed 
inside of the shell, however thin the shell may be; that is to say, 
the lines of force of the outside electric field do not penetrate 
into the metal but terminate at its surface. Therefore, the elec- 
tric charge on a metal body may be thought of as residing on 
the surface of the body. Figure 244 shows a hollow metal ball 
C placed between two charged bodies A and B. The presence 
of the ball C modifies the trend of the lines of force as may be 
seen by comparing Fig. 244 with Fig. 242, but the lines of force 
do not penetrate to the interior of the ball C. The interior of a 
metal shell is entirely screened from outside electric field. ^ This is 
an experimental fact. An electric field may be detected by its 
action upon a very light body like a suspended tooth-pick or a 
suspended pith ball. No evidence of electrical field can be 
detected inside of the ball C by such a device. 

Mechanical analogue of electrical screening. — Consider a mass 
of steel B, Fig. 245, which Is entirely separated from a sur- 
rounding mass of steel by an empty space eee. Stress and dis- 
tortion of the surrounding steel cannot affect B in any way, and 

* The screening is not complete while an electric field is changing rapidly. 



ELECTRIC CHARGE AND THE CONDENSER. 



311 



conversely stress and distortion of B cannot affect the sur- 
rounding steel, because the empty space is incapable of trans- 
mitting stress. This empty space, in its behavior towards 
mechanical stress, is analogous to a metal (or any electrical 





Fig. 244. 



Fig. 245. 



conductor) in its behavior towards electrical stress (electrical 
field). 

218. A charged conductor shares its charge with another con- 
ductor with which it is brought into contact. — A brass ball with a 
glass handle may be charged by touching it to one terminal of an 
influence machine, and if the brass ball is brought near to a sus- 
pended pith ball, as shown in Fig. 243, the charge on the brass 
ball will be indicated by the behavior of the pith ball. 

A brass ball A is charged by touching it to a terminal of an influ- 
ence machine, the ball A is then touched to another brass ball B {A 
and B both have glass handles), then both A and B are found 
to be charged. The charged ball A has shared its charge with 
ball B. The original charge on ball A is represented in Fig. 246, 
which shows the lines of force emanating from ball A; and the 
lines of force in Fig. 247 show how the charge originally on A 
has spread over A and B. See Art. 215. 

219. Giving up of entire charge by one body to another. — 



312 



ELECTRICITY AND MAGNETISM. 



Figure 248 shows a charged ball and an insulated metal can. If 
the charged ball is placed inside of the can, brought into contact 
with the inner wall of the can, and then removed, it will give up 





Fig. 246. 
Charge on single ball. 



Fig. 247. 
Charge shared by two balls. 



its entire charge to the can. This is true whatever charge the 
can may have to begin with. This giving up of the entire charge 
on one body to another, as described, is an essential feature in 
the action of the electric doubler and of the influence machine 
(see Arts. 221 and 223). 

ailk threadi 




can 



glass 



Fig. 248. 

The charged ball B is lowered into the can, and the can is 
closed by the metal lid as indicated in Figs. 249 and 250. As the 



ELECTRIC CHARGE AND THE CONDENSER. 



313 



charged ball Is lowered Into the can each line of force that 
emanates from the ball is cut in two, as it were, by the wall of the 



silk 
thread 





Fig. 249. 



Fig. 250. 



can so that when the ball Is entirely enclosed by the can as many 
lines of force emanate from the external surface of the can as 
from the ball B, as shown in Fig. 250. 





Fig. 251. 



Fig. 252. 



After the ball B has been completely inclosed by the metal 
can, the distribution of the electric field outside of the can is 



314 



ELECTRICITY AND MAGNETISM. 



entirely independent of what takes place inside of the can, 
because the walls of the can act as a complete screen as explained 
in Art. 217. 

If the ball B is then brought into contact with the inner wall 
of the can, as shown in Figs. 251 and 252, the lines of force which 
emanate from the ball disappear, as shown in Fig. 252, and all 
of the charge originally on B is left on the outside surface of 
the can. The ball may then be removed from the can and it 
will be found to be without charge, all of its original charge will 
have been given to the can. This giving up of the entire charge 
by the ball takes place however great the initial charge on the 
can may be. 

220. Charging by influence. — Charging by influence is essen- 
tially the cutting of electric lines of force in two by a sheet of 
metal so that one face of the metal sheet is negatively charged 
where the lines of force come in upon it, and the other face of 
the metal sheet is positively charged where the lines of force 
emanate from it. Thus Fig. 253 shows two metal balls A and B 
which have been brought in between two oppositely charged 
bodies C and D. The lines of force from C converge upon Ay 
and spread out from B as shown. If ^ and B are now sepa- 




Fig. 253. 



r"=n 



rated from each other and withdrawn from the region between 
C and Dy then B will be left positively charged (lines of force 



ELECTRIC CHARGE AND THE CONDENSER. 



315 



emanating from it), and A will be left negatively charged 
(lines of force coming in upon it) ; and the charges on C and 
D will be the same as at the beginning. 

221. The electric doubler. — ^The charged bodies C and D in 
Fig. 253 are metal cans supported on insulating stands as shown 



: glass handles 



4- 
+ 

+ 

4- 



+ 
+ 

+ 





d-. - 



B 






-glass 



(==t 




glass 



Fig. 254. 



also in Fig. 254. The ball A may be made to give up its entire 
charge to D by being placed inside of D and brought into con- 
tact with D; and the ball B may be made to give up its entire 
charge to C in a similar manner. The two balls A and B 
may then be again charged by being brought into the positions 
shown in Figs. 253 and 254; and the charges on A and B may 
again be delivered to D and C as before ; and so on. In this 
way the two cans C and D may be charged to any degree 
whatever, starting with any initial charges however small, pro- 
vided the insulation is very good. If the insulation is poor the 
charges leak away rapidly. 

A very interesting and striking experiment is the following. 
Two thin slabs of pith are attached to the cans C and D as 
shown in Fig. 255. Then fifty or more repetitions of the above 
described operation will charge both C and D sufficiently to 
make the thin slabs of pith stand out nearly horizontally. The 
success of this experiment depends upon extremely good Insula- 
tion. The cans C and D should be supported upon freshly 



3i6 



ELECTRICITY AND MAGNETISM. 



scraped blocks of hard paraffine or of cast sulphur, and the handles 
of the balls A and B should be fused quartz tubes closed at one 
end. 



slab of 
pith 



c 



strip of metal 



aMp of metal 




slab of 
pith. 



Fig- 255- 

222. The gold leaf electroscope. — ^The essential features of the 
gold leaf electroscope are shown in Fig. 256. A metal rod R is 
supported in the top of a glass case cc by means of an insulating 
plug. This plug is preferably made of cast sulphur. A metal 
disk D is fixed to the upper end of the rod, and two strips of gold 
leaf are hung side by side from the lower end of the rod. The 
glass case cc serves to protect the gold leaves from draughts of 
air. The sides of cc should be lined with metal strips ff, and 
these strips should be connected to earth. When the disk, rod 
and leaves are charged, the leaves are pulled apart by the lines 
of force which emanate from the leaves and terminate on the 
strips //, as shown in Fig. 257. 

The behavior of a gold leaf electroscope is as follows: (i) 
When the electroscope has no initial charge, the gold leaves di- 
verge when a positively or negatively charged body is brought 
near to the disk D. (2) If the disk or rod is touched with the 
finger when, say, a positively charged body Is near D, then the 
disk and leaves are left with a negati ve charge when the positively 
charged body Is removed to a distance. This is the operation of 
charging by influence which is described in a general way in 



ELECTRIC CHARGE AND THE CONDENSER. 



317 



Art. 220. (3) When the disk and leaves have an initial charge, 
the divergence of the leaves is increased by bringing a body with 
a like charge near D, and the divergence of the leaves is decreased 
by bringing a body with an unlike charge near D. 



, A , 



Z) 




Fig. 256. 
The gold-leaf electroscope. 



Fig. 257. 



223. The Toepler-Holtz influence machine. — ^The action of 
the Toepler-Holtz machine is essentially like the action of the 
electric doubler as described in Art. 221, except that in the 




Fig. 258. 
The Toepler-Holtz Electric Machine. 



3i8 



ELECTRICITY AND MAGNETISM. 



Toepler-Holtz machine each can in Fig. 253 is made of two sepa- 
rate parts. Thus C and C in Fig. 259 together take the place 
of C in Fig. 253, and D and D' together take the place of D 
in Fig. 253. 

A general view of a Toepler-Holtz machine is shown in Fig. 258, 
and the essential features of the machine are shown in Fig. 259. 
Metal carriers ccc travel along the dotted line in the direction 
indicated by the arrows. In positions i and 4 these carriers 
are under the influence of the charged bodies C and D, and 
they touch the neutralizing rod so that one carrier is left with an 
excess of negative charge and the other carrier is left with an 
excess of positive charge as shown. When the carriers reach the 
positions 2 and 5 they are momentarily connected to the charged 
bodies or inductors C and D to which they give up a portion of 
their charges. The inductors C and D are thus kept charged. 
When the carriers reach the positions 3 and 6 they make momen- 




-^ 



Fig. 259. 



ELECTRIC CHARGE AND THE CONDENSER. 319 

tary contact with C and D' to which they give up nearly all 
of their charges. This action is repeated over and over again. 
When the two cans in Fig. 253 are discharged it takes a long 
time for the doubling action to bring them up to a highly charged 
condition again. This difficulty is obviated in the Toepler-Holtz 
machine, because the terminals TT of the machine are not 
connected to the inductors C and Dy and therefore the inductors 
C and D are not discharged when the terminals TT are con- 
nected together, but chey continue to exert a strong charging 
influence upon the metal carriers as they pass positions i and 4. 

224. The spark discharge and the corona discharge. — When 
the electromotive force between two fairly large metal balls 
A and B is sufficiently increased the dielectric between A 
and B breaks down in the form of a brilliant, sharply defined 
spark extending from A to B, When the electromotive force 
between two needle points or between two very fine parallel 
wires is sufficiently increased the electric break-down of the air 
shows itself as a diffused luminosity surrounding the points or 
as a diffused luminosity surrounding the fine wires. This form 
of discharge is known as the corona discharge. When the electro- 
motive force between two needle points or between two fine 
wires is increased far beyond the value sufficient to produce the 
corona, a sharply defined spark is formed from needle point to 
needle point or from wire to wire. 

225. The Cottrell process for the precipitation of dust and 
smoke particles. — A fine wire is stretched along the axis of a 
metal tube; a steady {not alternating) voltage high enough to 
produce corona discharge is connected between wire and tube. 
The result is that the particles of smoke or dust are deposited on 
the walls of the tube and to a very slight extent on the wire. 

The most satisfactory arrangement for demonstrating the 
Cottrell process is shown in Fig. 260. A very fine wire A A, 
supported by two insulating glass posts, is stretched through a 
large glass tube, and a piece of strap iron BB is laid along the 



320 



ELECTRICITY AND MAGNETISM. 



bottom of the tube as shown in the figure. The fine wire and 
iron strap are connected to the terminals of an influence machine, 
and the smoke to be cleared is blown in at one end of the tube. 
The effect is most strikingly shown by short-circuiting the in- 
fluence machine by means of a small metal rod, filling the tube 
with smoke, and then suddenly removing the short circuit. 



U 



glass tube 



B' 



m 



to electric 
machine 



Jt 



Fig. 260. 

The fine particles of dust or smoke become electrically charged 
under the influence of the corona, and the charged particles are 
then pulled towards one of the electrodes A A or BB, in the 
same way that the charged metal ball b in Fig. 241 is pulled 
towards A or B. 

226. The ozonizer. — The ozonizer is a device for converting 
oxygen into ozone. The essential features of the device are 

shown in Fig. 261. Two 
metal plate A metal plates A and B, with 

a glass plate between, are 

connected to the terminals, 

ab, of the high-voltage coil 

of a step-up transformer, a 

blast of air is blown through 

between the plates, and a 

portion of the oxygen of the 

air is converted into ozone. 

The high electromotive force (alternating, of course) between 

the metal plates would produce a spark discharge from metal 

plate to metal plate in the absence of the glass plate, but with 



glass plate 
S S 



\ metal plate B 

Fig. 261, 



*»j 



Essential features of ozonizer (a and 
h connected to high voltage supply of 
alternating current). 



ELECTRIC CHARGE AND THE CONDENSER. 32 1 

the glass plate in position the entire region SS is filled with the 
purplish glow of the corona discharge. 

Ordinary oxygen has two atoms in the molecule. In the corona 
discharge these molecules are split into oxygen atoms, some of 
these oxygen atoms recombine as triplets, and these are molecules 
of ozone. Ozone is a very powerful oxidizing agent and it acts 
as an antiseptic. 



22 



CHAPTER XVI. 

THE ATOMIC THEORY OF ELECTRICITY. 

227. Mechanical theory and atomic theory. — The distinction 
between the atomic theory and thermodynamics is disscused in 
Art. 79, and a very brief outUne of some of the more important 
features of the atomic theory of gases is given in Art. 228. The 
atomic theory also covers an important field in electricity and 
magnetism, and the object of this chapter is to barely touch upon 
this phase of the atomic theory. 

The study of electricity and magnetism as represented in 
the foregoing chapters is independent of any consideration of 
the nature of the physical action which leads to the production 
of electromotive force in a voltaic cell or dynamo; it is inde- 
pendent of any consideration of the nature of the physical action 
which constitutes an electric current in a wire; it is independent 
of any consideration of the nature of the disturbance which 
constitutes a magnetic field; and it is independent of any con- 
sideration of the nature of the disturbance or stress which 
constitutes an electric field. This kind of study of electricity 
and magnetism may very properly be called electro-mechanics. 

Simple mechanics is the study of ordinary bodies at rest or in 
visible motion, and one of the most important ideas in mechanics 
is the idea of force; but the science of mechanics is not concerned 
with, and indeed it sheds no light upon the question as to the 
physical nature of force. Thus, the science of mechanics is not 
concerned with the question as to the nature of the action which 
takes place in a gas and causes the gas to exert a force on a piston; 
the science of mechanics is not concerned with the question as to 
the nature of the action which takes place in the material of a 
stretched wire causing the wire to exert a pull upon each of the 
two supports at its ends; the science of mechanics is not con- 

322 



THE ATOMIC THEORY OF ELECTRICITY. 323 

cerned with the nature of the action between the earth and a 
heavy weight which causes the earth to exert a force on the 
weight; the science of mechanics is not concerned with the 
nature of the action which takes place between two bodies 
which slide over each other and produces the force of friction. 
It is sufficient for the science of mechanics that these actions are 
what may be called states of permanency of the respective systems. 
For example, to say that a gas in a cylinder pushes with a force 
of 100 " pounds " on the piston thus compressing a spring is to 
refer to a state of affairs in which there is a clearly defined and 
maintained relationship between the condition of the spring and the 
condition of the gas. 

Similarly it is sufficient for the science of electro-mechanics 
that the physical actions that underlie electromotive force, 
electric current, magnetic field and electric field are what may 
be called states of permanency. Thus to say that a dynamo 
produces a current of 100 amperes through a circuit is to refer 
to a state of affairs in which there is a definite and maintained 
relationship between the dynamo and the circuit, the dynamo 
delivers energy at a certain rate and the circuit receives energy 
at a certain rate, and the circuit exhibits to a definite degree 
the various effects which are associated with what we call an 
electric current. 

The superficial character of the science of simple mechanics 
and of the science of electromechanics may be further exemplified 
as follows : (a) An engineer wishes to know the strength of a steel 
rod, and he finds by test that che rod is broken by a tension of 
'120,000 "pounds-"; but the exact character of the action which 
takes place in the steel when it is placed under tension is not a 
matter for consideration, neither does the engineer need to con- 
sider (as a part of his test) the action which goes on in the • 
furnace of the boiler that supplies steam to the engine that drives 
the dynamo that supplies current to the electric motor that 
drives his testing machine ! {b) An engineer wishes to know the 
"strength" of a glass insulator which he is to use to support 



324 ELECTRICITY AND MAGNETISM. 

a wire on a pole-line, and he finds by test that the insulator is 
broken down or punctured by an electromotive force of 95,000 
volts; but the exact character of the action which takes place 
in the glass when it is subjected to the electromotive force is 
not a matter for consideration, nor is it necessary to consider 
the changes which take place in the battery, for example, which 
may be used to produce the given voltage. 

Simple mechanics is concerned with the correlation of what 
may be called lump effects, such as the relationship between the 
size of a beam and the load it can carry, the size of a fly wheel 
and the work it can do when stopped, the thickness and diameter 
of a boiler shell and the pressure it can stand, the size of a sub- 
merged body and the buoyant force which acts upon it, the size 
and shape of the air column in an organ pipe and its number of 
vibrations per second, the thickness of a glass plate and the 
electromotive force it can stand, and so forth. 

The atomic theory, on the other hand, involves the develop- 
ment of more or less hypothetical conceptions of the minute 
details of physical action. The phenomena of chemical action 
and those physical phenomena which have to do with the minute 
details of physical processes have been studied heretofore almost 
solely on the basis of the atomic theory. 

A few years ago investigators used the terms atom and molecule 
reluctantly, because these terms were so largely hypothetical; 
but new experimental methods now enable the investigator to 
deal with actual individual atoms in the laboratory, and the old 
reluctance has largely disappeared. It must be remembered, 
however, that the electron and the atom and the molecule are 
still ideas In the use which the investigator makes of them; 
and the most hopeless state of mind that it is possible for the 
student of physics to get into is that which, for example, accepts 
the electric current as NOTHING BUT a flock of negatively 
charged specks butting blindly along in a wire.* In the old days 

* This is indeed the atomic conception of the electric current and it seems 
likely to lead to important and far reaching results. 



THE ATOMIC THEORY OF ELECTRICITY. 325 

many a student of physics, and we are now speaking of grown-up 
students, was so pleased to get Newton's second law of motion 
into his head (for it, too, is an idea) that he would say "Force, 
what is it? It is NOTHING BUT the product of mass and 
acceleration." Now It Is really a great achievement to get a 
precise Idea into one's head, and yet the complacency of the 
"Nothing But-er" Is a thing to be avoided. 

The reluctance of the Investigator to use hypothetical terms 
in speaking and especially In writing comes from the widespread 
misuse of such things in undisciplined and meaningless specula- 
tion. But no investigator ever hesitates to use such things in 
his work of investigation, for, as Helmholtz says, "It is a great 
help towards the sure understanding of all abstractions if one 
seeks to make the most concrete possible pictures of them, even 
when these pictures bring In many an assumption that Is not 
essential and necessary." 

"Our method," says Bacon, "is to dwell among things soberly, 
without abstracting or setting the mind farther from them than 
makes their images meet, and the capital precept for the whole 
undertaking Is that the eye of the mind be never taken off from 
things themselves, but receive their images as they truly are, 
and God forbid that we should offer the dreams of fancy for a 
model of the world." 

228. The atomic theory of gases.* — Imagine a large number of 
very small particles (molecules) moving about in a closed vessel. 
Imagine these particles (a) to have the property of rebounding 
with undiminished velocity when they strike the walls of the 
vessel, (Jb) to be so small as seldom to collide with each other, 
and {c) to exert no perceptible attraction or repulsion on each 

* An interesting simple discussion of the kinetic theory of gases, including van 
der Waal's theory, is given in Edser's Heat for Advanced Students, pages 287-314, 
The kinetic theory of gases forms one of the most important parts of mathematical 
physics. A good elementary treatise on the subject is Boynton's Kinetic Theory, 
The Macmillan Co., New York, 1904. See also Boltzman's Vorlesungen uber 
Gas Theorie. See also Planck, Acht Vorlesungen Uber theoretische Physik (Columbia 
University Lectures), Leipzig, 191 o. 



326 ELECTRICITY AND MAGNETISM. 

other. Such a system of particles would exhibit all the properties 
of an ideal gas (a gas which conforms exactly to Boyle's law and 
does no t have its temperature changed by free expansion) . There- 
fore a gas is thought of as if it were such a system of particles 
or molecules. 

Let p be the pressure of a gas, v its volume (the volume of 
the containing vessel), N the total number of molecules in the 
vessel, n the number of molecules per unit volume (= iV/z;), 
and m the mass of each molecule. The kinetic energy of the 
system of particles is constant, since the particles rebound from 
the walls with undiininished velocity. Therefore the average 
kinetic energy per molecule, namely, ^wco^, has a definite constant 
value, co^ being the average value of the squares of the velocities 
of the molecules. Then 

p = \nmoP' (i) 

or, since n = N/v, 

pv = \Nmcjp- (ii) 

Proof. — The square of the velocity of a given particle is equa} 
to the sum of the squares of the x, y and z components of its 
velocity. Therefore the sum of the squares of the velocities of 
all the particles is equal to the sum of the squares of all the 
x-components, plus the sum of the squares of all the T-com- 
ponents, plus the sum of the squares of all the z-components. 
The particles move at random in all directions, so that the sum 
of the squares of the :)c-components, of the j'-components, and 
of the z-components are equal each to each. Therefore The sum 
of the squares, Nco^, of all the velocities is equal to three times the 
sum of the squares of the x-components . Let us refer to this 
statement as (iii). 

Imagine the containing vessel to consist of two parallel walls, 
of area q, distant d from each other, perpendicular to the x-axis 
of reference, and between which the gas is confined. Only the 
x-components of the molecular velocities contribute, by impact, 
to the pressure on these walls, so that the y and z components 
may be ignored. Consider a single particle, the x-component of 



THE ATOMIC THEORY OF ELECTRICITY. 327 

whose velocity is a. This particle strikes first one wall and then 
the other, traveling back and forth a/2d times per second. 
At each impact the velocity of the particle changes by 2a, that 
is, from + a to — a, or the momentum* of the particle changes 
by 2am. Therefore momentum is lost on each wall by the 
impact of this particle at the average rate 2am X a/2d, or 
ma^/d, which is the average force exerted on the wall by this 
particle. That is, the force on one wall, due to one particle, is 
equal to m/d times the square of its x-velocity component. 
Therefore the total force F, exerted on the wall by all the 
particles, is equal to m/d times the sum of the squares of their 
:x:- velocity components. Therefore F = ^o^^N'm/d; see (iii) 
above; whence, dividing by q and putting qd = v, we have 
F/q = p = \Nmo2^lv. 

The kinetic theory of gases is very important as furnishing a clear 
conception of what constitutes thermal equilibrium of a gas, as 
furnishing a rational basis for Boyle's Law, Gay Lussac's Law, 
etc., and as enabling one to form clear mental pictures of various 
gas phenomena, as follows. 

Thermal equilibrium of a gas. — ^When a gas is in thermal 
equilibrium, the erratic movements of its molecules are such 
that on the average there is the same number of molecules in 
each unit of volume of the gas and the same average molecular 
velocity in the neighborhood of each point in the gas. 

Heating of a gas. — When the walls of a containing vessel are 
heated the molecules of the enclosed gas rebound with increased 
velocity when they strike the walls and the temperature of the 
gas rises. When the walls of the containing vessel are cooled, 
the molecules of the gas rebound with diminished velocity when 
they strike the walls and the temperature of the gas falls. 

* The momentum of a moving body is M =mv, where m is the mass of the 
body and v is its velocity. If :; changes M must change m times as fast. 
Therefore the rate of change of M is equal to ma, where a (the acceleration 
of the body) is the rate of change of v. But ma is equal to the unbalanced 
force acting on the body according to equation (i) of Art' 6. Therefore the un- 
balanced force which acts on a body is equal to the rate of change of the 
momentum of the body. 



328 ELECTRICITY AND MAGNETISM. 

Heating of a gas by compression. — Cooling of a gas by expan- 
sion. When a gas is compressed under a piston in a cylinder, 
the particles of the gas rebound from the inwardly moving piston 
with unchanged velocity relative to the piston, but with increased 
actual velocity, and the temperature of the gas rises. When a 
gas is expanded under a receding piston in a cylinder, the particles 
of the gas rebound from the receding piston with diminished 
actual velocity and the temperature of the gas falls. 

Boyle's law and Gay Lussac's law. — If we assume the absolute 
temperature of a gas to be proportional to the average kinetic 
energy per molecule, that is, if we assume T to be proportional 
to Jwco^, we may write constant X M X T for \Nmoy^ in 
equation (ii), and this equation then becomes 

pv = MRT 

which is the same as equation (46) of Art. 75, in which i? is a 
constant. On the basis, therefore, of the above assumption as 
to the relation between absolute temperature and average kinetic 
energy per molecule, the kinetic theory of gases is found to be in 
entire accord with Boyle's Law and Gay Lussac's Law. 

229. Electrons and ions. — When a charged gold leaf electro- 
scope is left standing it soon loses its charge and the leaves fall 
together. This loss of charge has long been known to be due 
in part to a leakage of the electricity through the surrounding air 
although usually it is due mostly to a leakage of electricity 
along the insulating supports. That is to say, air conducts 
electricity to some extent. There are a number of influences 
which cause the air (or any gas) to become a fairly good electrical 
conductor. Thus gas which Is drawn from the neighborhood of a 
flame or from the neighborhood of glowing metal or carbon is a 
fairly good conductor; gas which has been drawn from a region 
through which an electric discharge has recently passed is a 
fairly good conductor; a gas becomes a fairly good conductor 
under the action of Roentgen rays or under the action of the 
radiation from radioactive substances. The conductivity which 



THE ATOMIC THEORY OF ELECTRICITY. 329 

is imparted to a gas by these various agencies may be destroyed 
by filtering the gas through glass wool or by placing the gas for 
a few moments between electrically charged metal plates. This 
effect of filtration seems to shov\^ that the conductivity of the 
gas is due to something which is mixed with the gas, and the 
effect of the electric field (between the two charged plates) seems 
to show that this something is charged with electricity and is 
dragged out of the gas by the electric field. From some such 
considerations as these the hypothesis was originated that the 
electrical conductivity of a gas Is due to electrically charged 
particles floating around In the gas. These particles are called 
ions, and the process by which a gas is made into a conductor is 
called ionization; thus air is ionized in the corona discharge. 
See Arts, 224, 225 and 226. This hypothesis has been used 
extensively and with remarkable success in the study of electrical 
discharge through gases and in the study of radio-activity. 

The electron is a negatively charged particle of which the mass 
is about 1/800 of the mass of a hydrogen atom. The cathode 
rays, which are described later, are electrons thrown off from the 
cathode of the Crookes tube at high velocity, the jS-rays from a 
radio-active substance such as uranium are electrons which are 
expelled at extremely high velocity from the atoms of the sub- 
stance. 

An atom is supposed to be a kind of central nucleus with a 
number of electrons revolving round it, very much like the solar 
system which consists of the sun with the planets revolving around 
it. 

A molecule is a group of atoms clinging together. 

An ordinary atom or molecule is electrically neutral, that is, 
it has no apparent charge of electricity, the negative charges of 
the revolving electrons being offset by a positive charge In the 
nucleus. But when the molecules of a gas are broken up by 
chemical action, by the effect of high temperature, by Roentgen 
rays, or by the corona discharge the pieces are electrically charged, 
and these pieces may be electrons or simple ions. 



330 ELECTRICITY AND MAGNETISM. 

A simple ion is an atom from which a negatively charged 
electron has been detached thus leaving the remainder of the 
atom (the simple ion) positively charged; or a simple ion may 
be a neutral atom to which an electron has become attached 
thus giving the atom (the simple ion) a negative charge. 

The compound ion is an aggregate of two or more atoms or 
molecules with an extra electron attached to it thus giving it a 
negative charge or from which a negatively charged electron has 
been detached thus leaving the aggregate positively charged. 
In highly rarefied gases electrons and simple ions, only, exist; 
whereas compound ions may exist in gases at moderate or high 
pressure. 

230. Ionization by the electric field. — According to the kinetic 
theory of gases, a molecule of a gas travels on the average a 
certain distance between successive collisions with neighboring 
molecules. This distance is called the mean free path of the 
molecule. The mean free path of an electron in a gas is about 
4 V 2 times as great as the mean free path of the molecule of the 
gas,* because of the very small size of the electron, whereas 
the mean free path of an ion is equal to or ev^en less than the 
mean free path of the molecule of the gas. When a gas is sub- 
jected to an electric field by being placed between two oppositely 
charged metal plates, a certain amount of energy is imparted 
by the electric field to the electrons between successive collisions, 
and a much smaller amount of energy is imparted to the ions 
between successive collisions (because of their shorter mean free 
path). If the energy imparted to an electron between successive 
collisions exceeds a certain value, the electron is able to ionize 
an atom of the gas at its next collision, and when an atom of the 
gas is thus ionized a new electron and a new ion are produced. 
Similarly if the energy imparted to an ion between successive 
collisions exceeds a certain value, the ion is able to ionize an 

* The discussion of the electron theory in this chapter is necessarily very brief. 
The student who is interested in the subject should read Sir J. J. Thomson's Con- 
ductivity of Electricity through Gases, and Sir Ernest Rutherford's Radio-active 
Transformations. 



THE ATOMIC THEORY OF ELECTRICITY. 33I 

atom of the gas at its next collision thus producing a new ion 
and a new electron. Experiment shows that an electron must 
fall freely through a difference of potential of about 30 volts 
in order to receive enough energy to enable it to ionize an atom of 
oxygen or nitrogen, and an ion must fall freely through a differ- 
ence of potential of about 440 volts in order to receive enough 
energy to enable it to ionize an atom of oxygen or nitrogen. ' 

231. The electric spark in a gas. — ^When a gas is subjected to 
an electric field of which the intensity is sufficient to cause both* 
the electrons and the ions to ionize the gas, an extremely rapid 
and unlimited increase in the number of electrons and ions 
takes place, and the result is the production of an electric spark. 
The mean free path of the positive ions in a gas is inversely pro- 
portional to the pressure of the gas, according to the kinetic 
theory of gases, so that the electric strength of a gas should be 
proportional to the pressure. This is, in fact, the case. Thus, 
the dielectric strength of air at normal atmospheric pressure is 
about 32,000 volts per centimeter, at a pressure of 10 atmos- 
pheres the strength is about 320,000 volts per centimeter, and 
at a pressure of o.i atmOvSphere the dielectric strength is about 
3,200 volts per centimeter. The dielectric strength of air reaches 
a minimum, however, at a pressure of about 2 millimeters of 
mercury, and increases when the pressure falls below this value. 
An electromotive force, sufficient to produce a spark 1/8 of an 
inch long in air at atmospheric pressure, will produce a discharge 
through 18 or 20 inches of air at 2 millimeters pressure. 

The idea of dielectric strength is based on the assumption 
that the electromotive force required to produce a discharge is 

* When the intensity of an electric field is sufficient to cause only the electrons 
to ionize the gas, then all of the electrons form new ions and new electrons as they 
travel towards the positive electrode, but when they reach the positive electrode 
the action ceases except for the occasional formation of a stray electron by outside 
influences. When the electric field is sufficiently intense to cause electrons and 
positive ions both to produce ionization, then new ions and electrons are formed 
everywhere between the electrodes, and the number of free ions and electrons in- 
creases indefinitely. It is a well-known fact that an electric field must continue 
to act for an appreciable time before a spark is produced. 



332 



ELECTRICITY AND MAGNETISM. 




Fig. 262. 



proportional to the length of the spark, so that the quotient, volts 
divided by spark length, may be a constant. This is only 
approximately true in gases under moderate or high pressure,* 
and when the. pressure is very low a greater electromotive force 
is required to strike across a short gap than is required to strike 
across a long gap. This curious behavior of gas at low pressure 

is illustrated by a famous 
experiment due to Hittorf. 
Two electrodes were sealed 
into the walls of two glass 
bulbs and the tips of the 
electrodes were one milli- 
meter apart, as shown in 
Fig. 262. The two bulbs 
were connected together by 
a spiral tube 375 centi- 
meters long, and, when the pressure of the gas in the bulbs was 
reduced to a very low value, the discharge took place through 
the long tube and not across the one millimeter gap space between 
the points of the electrodes.! 

232. The Geissler tube and the Crookes tube. — The discharge 
of electricity through gases at low pressures is usually studied by 
means of a glass bulb through the walls of which are sealed 
platinum wires which terminate in metal plates called electrodes. 
The current enters at one electrode, the anode, and passes out at 
the other electrode, the cathode. This bulb, which is called a 
vacuum tube, is filled with the gas to be studied and the pressure 
is reduced to any desired value by exhausting the tube by means 
of an air pump. 

Before exhaustion the discharge through the tube is in the 
form of a sharply-defined spark similar to the spark in the open 
air. When the pressure of the gas in the bulb has been reduced 
to a few centimeters of mercury, the spark begins to be nebulous, 

* And, of course, between fiat electrodes. See Art. 209. 

t This behavior of a gas at low pressure is fully explained by the atomic theory. 
See J. J. Thomson's Conduction of Electricity through Gases, pages 430-527. 



THE ATOMIC THEORY OF ELECTRICITY. 



333 



and a continued reduction of pressure causes the luminosity 
ultimately to fill the entire tube. When the pressure has been 
reduced to a few millimeters of mercury the dis- 
charge presents the following features, as shown 
in Fig. 263. There is a thin layer of luminosity 
spread over the surface of the cathode C, and be- 
yond this there is a comparatively dark space D 
called the Crookes dark space, the width of which 
depends upon the pressure of the gas-, increasing 
as the pressure of the gas diminishes. This 
Crookes dark space extends to a boundary of the 
negative glow N. This boundary is approximately 
a surface traced out by lines of constant length 
drawn normally to the surface of the cathode. 
Beyond the negative glow is another comparatively 
dark region F which is called the Faraday dark 
space. Beyond the Faraday dark space is a 
luminous column P, called the positive column, 
extending to the anode A. This positive col- 
umn usually exhibits alternations of bright and 
dark spaces which are called striations. The effects 
here described are exhibited at their best in a vacuum tube in 
which the pressure has been reduced to a few millimeters of 
mercury. Such a vacuum tube is called a Geissler tube. 

When the exhaustion of the vacuum tube is carried further, 
the dark space which surrounds the cathode (the Crookes dark 
space) expands until it fills the entire tube. The glass walls of 
the tube then show a yellowish-green or blue luminescence 
(according as the tube is made of soda glass or lead glass) , and 
a slight negative glow may remain in portions of the tube remote 
from the cathode. These effects, which were first studied by 
Crookes in England and by Pliicker and Hittorf in Germany, 
are exhibited at their best in a vacuum tube in which the pres- 
sure has been reduced to a few thousandths of a millimeter of 
mercury. Such a vacuum tube is called a Crookes tube. 




Fig. 263. 



334 ELECTRICITY AND MAGNETISM. 

233. Cathode rays and canal rays. — In order that a steady 
discharge may flow through a vacuum tube, it is necessary that 
the electric field intensity reach a value sufficient to impart to 
the positive ions enough energy between collisions to enable them 
to ionize the molecules of the residual gas, because if the electrons 
(negative ions), only, produce ionization, the discharge through 
the tube ceases very soon after all of the negative ions have moved 
across to the neighborhood of the anode. In fact, ionization 
by positive ions must take place in the neighborhood of the 
cathode,* and it is this necessity which gives rise to the Crookes 
dark space. The action which takes place in the Crookes dark 
space is as follows: Electrons (negative ions) are thrown off from 
the cathode at very high velocity by the intense electric field in 
the Crookes dark space, very energetic ionization takes place 
in the negative glow N, Fig. 263, and the positive ions that are 
produced in this region attain sufficient velocity in traveling 
towards the cathode to enable them to ionize the gas in the 
immediate neighborhood of the cathode. That is, ionization by 
positive ions takes place in the faint glow which covers the 
cathode. The mutual dependence of the ionization which takes 
place in the negative glow and the ionization which takes place 
in the faint luminosity in the immediate neighborhood of the 
cathode is shown by placing a small obstacle in the Crookes 
dark space. This obstacle screens a portion of the cathode sur- 
face from bombardment by the positive ions which move from 
the negative glow towards the cathode so that the region so 
shaded is free from even faint luminosity because ionization does 
not take place there. In the same way the obstacle also screens 
a certain portion of the negative glow from bombardment by the 
electrons which are thrown from the cathode and this portion of 
the negative glow ceases to exist because ionization is no longer 
produced there. That is to say, the obstacle casts a shadow on 
the cathode and it also casts a shadow into the negative glow. 

* A detailed discussion of this matter may be found in J. J. Thomson's Con- 
duction of Electricity through Gases, pages 529-603. 



THE ATOMIC THEORY OF ELECTRICITY. 335 

The electrons which are thrown off from the cathode attain a 
very high velocity as they travel across the Crookes dark space, 
and many of these electrons travel In approximately straight 
lines until they strike the walls of the tube. These high velocity 
electrons constitute what are called cathode rays. The cathode 
rays are faintly visible throughout the tube because of occasional 
collisions with molecules of the gas, and where they strike the 
glass walls of the tube they produce brilliant luminescence. 

An object of any kind placed in the Crookes tube casts a sharp 
shadow upon the wall of the tube, as shown in Fig. 264. The 
wall of the tube shows a brilliant luminescence everywhere except 
where It Is screened by the obstacle from bombardment by the 
cathode rays. 

When the cathode has a small hole through It, the positive ions 
which move towards the 
cathode from the nega- 
tive glow pass through 
this hole In the form of 
a stream of rays which 
Is made visible by the 
luminosity which accom- 
panies the collisions of 
the positive Ions with the 
molecules of the gas. 

Such a stream of positive ions constitutes what has been called 
the canal rays. 

234. Magnetic deflection of cathode rays and canal rays. — 

A moving charged body Is equivalent to an electric current, and 
when a charged body moves across a magnetic field the magnetic 
field pushes sidewlse upon the charged body and causes the 
charged body to describe a curved path. Therefore, If the 
cathode rays and canal rays are rapidly moving charged particles 
they should be deflected by a magnetic field. This is, Indeed, 
found to be the case. 

The magnetic deflection of the cathode rays is easily shown 




336 ELECTRICITY AND MAGNETISM. 

by placing a horse-shoe magnet with its poles placed on opposite 
sides of the tube shown in Fig. 264. The shadow of the cross is 
thrown up or down according to the arrangement of the magnet. 
The magnetic deflection of the canal rays is very slight; a very 
strong magnetic field is necessary to produce a perceptible deflec- 
tion. The direction of the magnetic deflection of the cathode 
rays shows that these rays are negatively charged particles, and 
the direction of the magnetic deflection of the canal rays shows 
that these rays are positively charged particles. The magnitude 
of the deflection of the cathode rays shows that the mass of the 
cathode particles (electrons) is very small and that their velocity 
is very great. The magnitude of the deflection of the canal rays 
shows that the mass of the canal ray particles is relatively great 
and that their velocity is less than the velocity of the cathode 
rays. 

An object upon which the cathode rays* impinge is heated, it 
may be, to a very high temperature. Many substances, how- 
ever, emit light without being made perceptibly hot when sub- 
jected to bombardment by the cathode rays. Such substances 
are said to be luminescent. For example, lead sulphate emits 
a deep violet light, zinc sulphate emits white light, magnesium 
sulphate, with a slight admixture of manganese sulphate, emits 
a deep red light under the action of cathode rays. 

The cathode rays pass quite readily through thin metal plates, 
especially through thin plates of aluminum. By using a Crookes 
tube of which a portion of the wall is made of thin sheet alumi- 
num, the cathode rays may be made to pass through into the 
outside air. The properties of cathode rays thus obtained in the 
air were first studied by Lenard who found that they produce a 
very high degree of ionization of the air making it a fairly good 
electrical conductor. Lenard found the cathode rays capable of 
traversing from 10 to 20 centimeters of air at atmospheric pres- 

* The cathode rays produce effects which are practically important and which 
can be easily observed. The effects of the canal rays, however, are so slight as to 
be scarcely perceptible even under the most favorable conditions. Therefore 
further discussion of the canal rays is not warranted in this brief outline. 



THE ATOMIC THEORY OF ELECTRICITY. 337 

sure, he found them capable of exciting luminescence, and he 
found them capable of affecting a sensitive photographic plate. 

235. The Roentgen rays. — Objects upon which the cathode 
rays impinge, not only become heated and luminescent as de- 
scribed above, but they emit a type of radiant energy which was 
discovered by Roentgen in 1894. Roentgen rays are of the same 
physical nature as light rays, that is, they consist of waves in 
the ether, and they are related to light waves very much as 
fine ripples are related to the long-wave-length ocean swell. 
Helmholtz pointed out in 189 1 that "light waves" of extremely 
short wave-length would have certain properties, the properties, 
in fact, which are exhibited by Roentgen rays, as follows: These 
rays are not reflected in a regular way by the surface of a mirror, 
and they are not refracted by a lens. They pass through all 
substances, subject to a certain amount of absorption which is 
greater the greater the density of the substance, and subject to 
a certain amount of diffused scattering. The Roentgen rays 
affect an ordinary photographic plate and they have a powerful 
ionizing action on gases. 

The fiuoroscope. — Many substances such as barium platino- 
cyanide and calcium tungstate become luminescent under the 
action of Roentgen rays. This effect is utilized in the fluoroscope 
which consists of a cardboard screen covered with a layer of 
barium platinocyanide. When the Roentgen ray shadow of an 
object, such as the hand, falls on this screen the shadow becomes 
visible; where the Roentgen rays have been greatly reduced in 
intensity by the bones of the hand the screen remains dark, 
where the Roentgen rays have been slightly reduced in intensity 
by the flesh the screen is moderately luminous, and where the 
rays have not been reduced at all in intensity the screen is highly 
luminous. The Roentgen ray shadow of an object may be ren- 
dered visible by allowing it to fall upon a photographic plate and 
developing the plate as in ordinary photography. 

236. The emission of electrons by a hot metal.* — The exten- 

* This subject has been investigated with great care by Professor O. W. Richard- 
23 



338 



ELECTRICITY AND MAGNETISM. 



sive investigations of Professor Richardson show that a hot metal 
surface emits electrons in great numbers, especially if the tem- 
perature of the metal is very high, and these electrons leave the 
metal at very low velocities. One may indeed think of the 
emitted electrons as forming a comparatively stagnant cloud 
in the neighborhood of the hot metal surface. 

The emission of electrons by a hot metal surface is made use 
in three important practical appliances, namely, (a) The Coolidge 
X-ray tube or Roentgen-ray tube, (b) The vacuum tube rectifier 
or kenetron, and (c) The telephone repeater or amplifier. All 
of these devices depend upon the carrying of current by the 
negatively charged electrons in the absence of positively charged 
ions, and consequently the stream of electrons which carries the 
current must be in an extremely high vacuum; if an appreciable 
amount of gas is present the molecules of the gas will be ionized 
thus producing positively charged ions. 

The essential features of the Coolidge tube are shown in Fig. 
265. A high-voltage supply maintains a very intense electric 




direction of 
current 



Fig. 265. 

field between a metal cathode CC and a massive tungsten anode 
AA. A fine filament of tungsten wire, /, like an incandescent 
lamp filament, is connected to and forms part of the cathode CC, 

son, whose papers have been published chiefly in the Philosophical Magazine from 
1904-1916. 



THE ATOMIC THEORY OF ELECTRICITY. 339 

and this filament Is kept at any desired high temperature by the 
current from a low- voltage battery B. The electrons which 
are given off by the hot filament / are pulled across from C 
to A by the electric field in a slightly converging stream and 
they impinge on a small spot on the face of the tungsten anode. 
The stream of electrons, which is indicated by the fine dots in 
Fig. 265, constitutes what is called the cathode rays, and the fine 
radiating lines represent the X-rays radiating from the spot 
where the cathode stream strikes the tungsten plate AA, 

With the Coolidge tube the current* through the tube can be 
controlled by changing the temperature of / thus changihg the 
number of electrons per second which are available for carrying 
negative charge from C to ^, and the supply voltage E can 
be high or low as you please. The higher the voltage E the more 
penetrating the X-rays, and the greater the current with a given 
voltage the more intense (the greater the energy value of) the 
X-rays. With the Coolidge tube these two qualities of the 
X-rays can be adjusted independently of each other. 

Current is carried through the Coolidge tube by the negative 
electrons, and negative electrons are supplied only at the plate 
CC (at the filament /).t Therefore current can flow through the 
Coolidge tube in one direction only, in the direction corresponding 
to the travel of negative charge from C to ^ , as indicated by 
the arrows ii. The Coolidge tube can, indeed, be operated by 
connecting C and A to the high-voltage coil of a step-up 
transformer. When the alternating voltage is in one direction a 
stream of electrons is carried across from C to A and X-rays 
are produced, but when the alternating voltage is reversed no 
current can flow. 

* With a given voltage the current is limited and, under ordinary conditions, 
partly determined by the distribution of negative charge on the electrons which 
are en route between C and A , and because of this space-charge effect the statement 
that intensity and penetration of the X-rays can be independently controlled is 
not strictly true, although practically it is true. 

t The plate A A may become very hot on account of the bombardment by the 
particles in the cathode stream, and if AA does thus become hot it gives off 
electrons. 



340 



ELECTRICITY AND MAGNETISM. 



This behavior of the Coolidge tube when it is operated by 
alternating voltage is utilized in the vacuum tube rectifier or 
kenetron. 

The telephone amplifier. — The telephone amplifier or repeater 
is a device whereby the very weak current "wave" from a distant 
telephone transmitter can control a new source of energy and 
reproduce the "wave" on a greatly intensified scale with all its 
characteristics, so that the "wave" has a new lease of life, as it 
were, and can travel over another long stretch of telephone line 
and actuate a distant telephone receiver. The essential features 
of the repeater are shown in Fig. 266. The two wires T come 
from the distant telephone transmitter, and the two wires 
R lead to the distant telephone receiver. To understand the 
action of the device we must consider the effect of the small 
battery A. Without this battery the "wave" from the distant 
transmitter would cause the voltage between C and D to vary 



4 



Ia-' 



C \D 



fine wire grid 



high vacuum 




i'l'l'l'l'l'l'l'l'-y6-o-oowo-^ 



LOAAiL 



^R 



Fig. 266. 



as represented in Fig. 267, but the constant added voltage of the 
battery raises the entire curve above the zero axis so that the 
voltage between C and D is always in the same direction (in 
the direction to drag a negatively charged electron from C to 
D) but varying in value as shown in Fig. 268. 

The space between D and E is normally empty, and the 
battery B is on open circuit; but the varying voltage between 
C and D drags electrons across from the hot electrode (where 



THE ATOMIC THEORY OF ELECTRICITY. 341 

the electrons are formed) to the grid C, and a certain fraction of 
these electrons shoot through the meshes of D and enter the 
region between D and E where they move across from D to 
E carrying current from the battery B. The number of elec- 
trons per second thus shooting through D is proportional to the 
voltage between C and D as represented in Fig. 268, and the 
current delivered by B is proportional to the number of electrons 
per second that become available in the region between D and 




Fig. 267. Fig. 268. 

E. Therefore the current from battery B rises and falls in a 
manner represented by the ordinates of the curve in Fig. 268, this 
fluctuating current flows through the primary P of a small 
transformer, and the secondary 5 gives out energy in the form 
of a reproduced and greatly intensified "wave" which actuates 
the distant telephone. 

The device which is shown in Fig. 266 is used extensively in 
wireless telegraphy under the name of the audion. 

237. Radio-activity.* — The chemical elements uranium, tho- 
rium, and radium and their compounds have the property of 
making a surrounding gas an electrical conductor. Thus, one 
ten-millionth of a gram of radium bromide which is left as a 
residue upon a metal plate by evaporating a small quantity of a 
dilute solution of radium bromide on the plate, causes a gold leaf 
electroscope to be discharged in a few seconds when the radium- 
covered plate is held near to the metal plate of the electroscope. 

* The student is referred to the following books for a full discussion of radio- 
activity: Radioactivity, by E. Rutherford, Cambridge, 1905 (second edition); 
Radioactivity, by Frederick Soddy, London, 1904; and Radioactive Transformations, 
by E. Rutherford, New York, 1906. See also Rutherford's Radioactive Substances 
and their Radiations, Cambridge, 1913. 



342 ELECTRICITY AND MAGNETISM. 

Uranium and thorium have the same effect but the divscharge 
which they produce is not so rapid unless a large quantity of 
material is employed. This property of these metals and of their 
compounds is called radio-activity, a name which originated 
because of the peculiar radiations which are given off by these 
substances and to which the discharging action is due. These 
radiations are of three distinct kinds, which are called the a-rays, 
the /5-rays, and the 7-rays, respectively. The 7-rays penetrate 
through a foot or more of solid metal or through many feet of air ; 
the j6-rays penetrate through a moderate thickness of a light 
metal, such as aluminum; whereas the a-rays are stopped by a 
very thin layer of aluminum or by a layer of air two or three 
inches in thickness. 

The a-rays consist of positive ions each about four times as 
massive as a hydrogen atom. These ions are projected from the 
radioactive substance at a velocity of about 20,000 miles per 
second, and each of them ionizes about 100,000 air molecules 
before it is brought to rest by repeated collision. After traveling 
two or three inches through the air, the velocity of these a-par- 
ticles is reduced to so low a value as to render them no longer 
perceptible by their ionizing effects. 

The /3-rays consist of electrons (negative ions) each about 1/800 
as massive as a hydrogen atom. These electrons are projected 
from the radio-active substance at a velocity which in some cases 
is nearly as great as the velocity of light (186,000 miles per 
second). The j3-particles also have the property of ionizing the 
gas through which they pass but not to so great an extent as the 
«-particles, and they travel several feet through the air before 
their velocity is reduced to so low a value as to render them no 
longer perceptible by their ionizing effects. 

The 7-rays are essentially like X-rays but of shorter wave- 
length and greater penetrating power. The 7-rays also have the 
power of ionizing a gas. 

The present hypothesis regarding radio-activity is that the 
atom of a substance is a system of excessively small negatively 



THE ATOMIC THEORY OF ELECTRICITY. 343 

charged particles, called electrons, revolving around a nucleus 
containing positive charge, the atom of each element being a 
characteristic group or system. These systems of electrons 
(atoms) are supposed to be to some extent unstable, and when 
instability occurs, the system (atom) collapses into a new con- 
figuration, and at the same time expels one or more positively or 
negatively charged particles which constitute the a-rays and 
the ^-rays. 



-^» ■ 



Let the reader consider well the intent of this brief chapter 
with its glib talk about atoms and alpha-particles, about' beta- 
particles with velocities exceeding ten million miles a minute, 
about Roentgen rays and gamma-rays with wave-lengths less 
than a ten-millionth of a millimeter, about radium atoms as 
unstable solar systems unthinkably small of which half the 
number in existence today will have exploded in 1300 years, and 
so on! Indeed all these astounding things and many more 
besides might have been included in the chapter if it were desir- 
able to dazzle the reader. But the chapter is sufficiently glib 
as it stands, and the worst thing about it is that by virtue of its 
glibness it may be interesting to many a reader. 

Science is FINDING OUT. If a healthy minded person 
takes an interest in science, he gets busy with his mathematics 
and haunts the laboratory; having no such interest, a man should 
exercise the great virtue of ignorance, holding all things in con- 
tempt which he does not really understand. And ignorance 
operating thus is the greater part of wisdom 1 



PART IV. 
THE THEORY OF LIGHT. 

Three good books for the beginner are Edser's Light for Students, Macmillan Sc 
Co., London, 1902; Preston's Theory of Light, MacMillan & Co., London, 1890; 
and Wood's Physical Optics, The Macmillan Co., New York, 1914. 

Advanced and Special Treatises. 

Lehrbuch der Optik, Paul Drude, Leipsig, 1900. This book has been translated 

into English by Professors Mann and Millikan of Chicago. 

A very interesting and semi-popular book is Light Waves and their Uses, by 
A. A. Michelson, University of Chicago Press, 1903. 

Winkelmann's Handhuch der theoretischen Physik, 2d edition, Vol. VI, contains a 
very full discussion of the theory of lenses and of optical instruments. 

Photographic Optics, Otto Lummer, translated by S. P. Thompson, Macmillan Sc 
Co., London, 1900, is an important book. 

Handhuch der Spectroskopie, Heinrich Kayser, 6 volumes, Leipsig, 1900- 191 2, is 
the most complete work on spectrum analysis and spectroscopy. Other good 
books are: 

Spectrum Analysis^ John Landauer, translated by J. Bishop Tingle, John Wiley & 
Sons, New York, 1907- 

Spectroscopy, E. C. C. Baly, Longmans, Green & Co., London, 1912. 

Text Book of Color, O. N. Rood, D. Appleton & Co., New York, 1881. 

An extremely interesting discussion having much to do with color effects is 
Helmholtz's popular lecture on The Relation of Optics to Painting, translated by 
E. Atkinson in the Second Series of Helmholtz's Popular Lectures, Longmans, Green 
& Co., London, 1903. Three lectures in the First Series of Helmholtz's Popular 
Lectures (translated by Dr. Pye-Smith, Longmans, Green & Co., 1873) On the 
Theory of Vision are of great interest. 
Handhuch der physiologischen Optik, H. von Helmholtz, 3d edition, 2 vols., Hamburg, 

1910-11, is a work of great importance. 

A very simple but fairly complete treatise on Photometry and Illumination is 
to be found on pages 86-183 of W. S. Franklin's Electric Lighting, The Macmillan 
Co., New York, 191 2. 

An important reference book is The Johns Hopkins University Lectures on 
Illuminating Engineering, Baltimore, 191 1. 



CHAPTER XVII. 

LIGHT AND SOUND DEFINED * VELOCITY OF TRANSMISSION. 

238. Sensory nerves. — The sensory nerves of the human body 
lead from regions near the surface of the body to the central 
organs of the nervous system. The outer ends of these nerves 
are exposed in such a way as to be excited or set into commo- 
tion by physical disturbances in the region surrounding the body, 
this commotion is transmitted to the central organs producing 
commotion there, and we experience what we call a sensation. 
The physical disturbance which excites the nerves is called a 
stimulus. 

239. Proper stimuli. End organs. Localization. — Each set of 
sensory nerves, such as the nerves of sight or the nerves of 
smell, is especially sensitive to a certain kind of disturbance, and 
the disturbance to which a given set of sensory nerves is espe- 
cially sensitive is called the proper stimulus of that set of nerves. 

A set of sensory nerves is rendered especially sensitive to its 
proper stimulus in two distinct ways, namely, {a) by being pro- 
vided with specialized end organs, and {h) by being located so as 
to be exposed to the proper stimulus but protected to a great 
extent from other stimuli. Thus, the nerves of sight terminate 
in minute organs, the so-called rods and cones, which are situated 
in the retina of the eye. These organs are sensitive to the dis- 
turbances which can reach them through the transparent humors 
of the eye, and they are to a great extent protected from all other 
physical disturbances. 

When a given set of sensory nerves is excited, the sensation 
which corresponds to the set of nerves is always produced no matter 
what the character of the stimulus may be. Thus, a sensation 
of light is always produced when the nerves of sight are excited, 

* It is convenient in this chapter to treat both light and sound. 

347 



I 



348 THE THEORY OF LIGHT. 

be the excitation caused by a severe mechanical shock, by an 
electric current passing through the eye, by abnormal conditions 
of blood circulation through the retina, or by the proper stimulus 
(the light which passes through the transparent humors of the 
eye. See next article). 

240. Light, the sensation ; light, the proper stimulus. — The sen- 
sation which is experienced when the nerves of sight are excited 
is called light. That physical disturbance which constitutes the 
proper stimulus of the nerves of sight is also called light. The 
most familiar property of the physical disturbance which is called 
light is that it can pass through many substances, such as glass 
and water; that is to say, we can "see through" such sub- 
stances, and they are said to be transparent. 

The study of light as a sensation belongs to the subject of 
psychology; the study of the physical disturbance which consti- 
tutes the proper stimulus of the nerves of sight belongs to the 
subject of physics. This treatise is devoted to the study of light, 
the proper stimulus. 

241. Nerves of hearing. Sound, the sensation: sound, the 
proper stimulus. — The nerves of hearing terminate in end organs 
which float in a watery fluid which is contained in a bone-walled 
cavity called the inner ear. These end organs are to a great 
extent protected by the massive bones of the head from all dis- 
turbances except vibratory movements of the air which reach 
them as follows: Figure 269 shows the essential features of the 
ear.* The cavity C which is filled with watery fluid constitutes 
the inner ear, dind it is provided with two "windows," the oval 
window and the round window R which are closed by very 
thin membranes. The middle ear M is an air-filled cavity which 
communicates with the mouth cavity through the Eustachian 

* This figure is not intended to show the actual structure but to illustrate the 
principles of action of the ear. No attempt has been made in this figure to show the 
cavity of the inner ear (the cochlea) in which the basilar membrane is situated, which 
contains the sense organs of musical pitch. The principles of action of the basilar 
membrane are explained in Chapter XXXI. A very full discussion of the structure 
and action of the ear may be found in Helmholtz's Tonempfindungen, pages 189-226 
of the English translation by Alexander J. Ellis (Longmans & Co.). 



LIGHT AND SOUND DEFINED. 



349 



outside 
air 




Fig. 269. 



tube E, and three small bones In the form of a chain bridge 
across from the ear drum D to the membrane which covers the 
oval window. The vibratory- 
motion of the outside air causes 
the ear drum D to vibrate. 
These vibrations are transmitted 
to the oval window by the chain 
of small bones, the vibration of 
the membrane of the oval win- 
dow causes the fluid in the inner 
ear to surge back and forth 
through the complicated chan- 
nels of the inner ear between the 

oval window and the round window, as indicated by the double- 
headed arrow in Fig. 269, and the end organs of the nerves of 
hearing are excited by this surging fluid. 

The sensation which is experienced when the nerves of hearing 
are excited is called sound. That physical disturbance which 
constitutes the proper stimulus of the nerves of hearing is also 
called sound. The study of sound, the sensation, belongs to the 
science of psychology. This treatise is devoted to the study of 
sound, the proper stimulus. 

242. Transmission of light and sound.^ — We have come by 
experience to associate distant objects with our sensations of 
light and sound — that is we see and hear distant objects — and 
the attempt to explain the evident connection between a distant 
object and the sensations of light and sound which it produces 
has led philosophers to assume the existence of a physical agent 
which we call light and which is transmitted to our eyes from a 
distant object; and to assume the existence of a physical agent 
which we call sound and which is transmitted to our ears from a 
distant object. 

243. The corpuscular theory of light.* — The phenomena of 

* A very interesting discussion of the corpuscular theory of light is to be found in 
Preston's Theory of Light, pages 15-21. Sir Isaac Newton was the greatest ex- 
ponent of the corpuscular theory. 



350 THE THEORY OF LIGHT. 

shadows and the obstruction of vision of a distant object by inter- 
vening objects show that Hght travels sensibly in straight lines. 
In accordance with this fact, it was the accepted theory, until 
long after the time of Sir Isaac Newton, that light consisted of 
particles or corpuscles which were thrown off from luminous 
bodies at great velocity, traveling in straight lines until reflected 
or stopped by objects upon which they impinged. This was 
called the corpuscular theory of light. 

244. The wave theory of light and sound.* — The most compre- 
hensive understanding of the phenomena of light and sound has 
been reached on the hypothesis that light and sound are wave- 
like disturbances which pass out in all directions from luminous 
bodies and from sounding bodies, respectively. No attempt will 
be made to lead up to this hypothesis by preliminary discussion, 
but its justification, in the reader's mind, will become more and 
more complete as he has occasion to use it. 

245. Transmitting media. The luminiferous ether. — The con- 
ception of light as a wave-like disturbance requires the assump- 
tion of a transmitting medium. The fact that light reaches us 
from the sun and stars across apparently empty space, and the 
fact that no known material substance is capable of transmitting 
waves with anything approaching the enormous velocity of light, 
necessitate the assumption of a special light-transmitting medium, 
the ether, which fills all space. 

Sound, on the other hand, cannot travel through a vacuum; 
the transmission of sound is, in fact, accomplished by a wave-like 
disturbance of air, or water, or other material medium. 

246. Velocity of sound. f — It is a familiar fact that sound 
requires a perceptible time to reach the ear from a sounding 

* See " The Wave Theory of Light," a collection of original memoirs by Huygens, 
Young and Fresnel, translated and arranged by Henry Crew. The Scientific 
Memoirs Series, edited by J. S. Ames. 

t The velocity of sound varies quite perceptibly with the loudness. See Encyclo- 
pedia Britannica, 9th edition, article Acoustics, sections 20 to 27. The velocity of 
sound varies greatly with the temperature of the air. This matter is discussed in the 
Britannica article. See also Poynting and Thomson's Text-Book of Physics, Volume 
on Sound, pages 16-31. 



LIGHT AND SOUND DEFINED. 351 

body. The firvSt attempt to measure accarately the velocity of 
sound was made by a committee of members of the French 
Academy of Sciences in 1738. The observers were placed at the 
Paris Observatory and at three distant stations visible from the 
observatory. Every ten minutes a cannon was fired at one of 
the stations, and the observers at the other stations noted the 
time intervals which elapsed between the flash and the sound of 
the cannon. The light flash was transmitted almost instan- 
taneously so that the observed time intervals were taken to be 
the time intervals required for the sound to travel over the 
measured distances between the stations. 

In 1822 this experiment was repeated at Paris in a slightly 
modified form. Two stations were selected, cannon were fired at 
these stations alternately, and the time intervals between flash 
and sound were observed as before. By firing at the two stations 
alternately the influence of the wind was eliminated. The dis- 
tance between the stations was 18,622 meters (about 10 miles). 
The observed time interval between flash and sound at one sta- 
tion was 54.84 seconds and at the other station it was 54.43 
seconds. These data gave 340.9 meters per second as the veloc- 
ity of sound ; allowing for the temperature of the air at the time 
the observations were taken, this corresponds to 331.2 meters 
per second at 0° C. More recent determinations of the velocity 
of sound give 331 meters per second for the velocity in dry air 
at 0° C. The velocity of sound in air depends upon the temper- 
ature, and it also depends upon humidity. 

The following table gives the velocity of sound in various sub- 
stances in terms of the velocity of sound in air: 

TABLE. 
Velocities of Sound in Different Media. 

Air Unity 

Iron 15.1 

Glass 15.3 

Water 4.3 

247. Velocity of light.* — ^A Danish astronomer, Roemer (1675), 

* For full account of the researches upon the velocity of light, see Preston's 
Theory of Light, Chapter XIX. 



352 THE THEORY OF LIGHT. 

was the first to show that Hght has a finite velocity. He found 
that the observed time of revolution of the satellites of Jupiter 
varies with the position of the earth in its orbit. When the 
earth is moving to^vards Jupiter the observed time of revolution 
of a satellite is less than the true time of revolution, and when 
the earth is moving away from Jupiter the observed time is 
greater than the true time; the true time of revolution being 
the mean of all the observed times daring one revolution of the 
earth in its orbit. The cause of the variation of the observed 
time of revolution of Jupiter's satellites is as follows: Suppose 
a light signal is flashed at equal intervals of time. A stationary 
observer sees these flashes separated by the true time interval 
between them. If, during the interval between two flashes, the 
observer moves towards or away from the source of the flashes, the 
interval between observed flashes will be less or more than the 
true interval, and the amount of this difference will be equal to 
the time required for light to pass over the distance that the 
observer has moved. 

A laboratory method for measuring the velocity of light was 
devised by Fizeau in 1849. This method was employed under 
more favorable conditions by Cornu in 1874. Another method 
was devised by Foucault in 1850. This method has been used 
by various observers, notably by Michelson in 1880. Newcomb 
in 1882 carried out Foucault's method. He found, from an 
extended series of observations, a velocity of 299,778,000 meters 
per second in air (299,860,000 meters per second in vacuum). 
This is perhaps the most reliable determination that has been 
made. 



CHAPTER XVIII. 

SOME USEFUL IDEAS OF WAVE MOTION. 

248. Wave pulses and wave trains. — ^When a stone is pitched 
into a pond a wave emanates from the place where the stone 
strikes. When a long stretched wire is struck sharply with a 
hammer, a single wave (a bend in the wire) travels along the wire 
in both directions from the point where the wire is struck. When 
a long steel rod is struck on the end with a haminer, a single 
wave (an endwise compression of the rod) travels along the rod. 
When an explosion takes place in the air, the firing of a gun for 
example, a single wave (a compression of the air) travels out- 

.wards from the explosion. Such isolated waves are called wave 
pulses. 

When a disturbance at a point in a medium is repeated in 
equal intervals of time the disturbance is said to be periodic. 
Such a disturbance sends out a succession of similar waves con- 
stituting what is called a wave train. 

The distance between similar parts of two successive waves 
of a train is called the wave-length of the train. If V is the 
velocity of travel of the waves and r the time which elapses 
between repetitions of the periodic disturbance which produces 
the wave train, then Vt is the distance traveled by one wave 
when the next succeeding wave is starting out. Therefore 

\= Vt (1) 

where X is the wave-length of the train. 

249. The wave front.— One of the most important ideas in 
conneccion with the wave theory of light is the Idea of the wave 
front. Every one is familiar with the fact that waves on the 
surface of a pond always resolve themselves into clearly defined 
ridges at a distance from the disturbance, however complicated 

24 353 



354 



THE THEORY OF LIGHT. 



the disturbance may be. Thus, when a handful of pebbles is 
thrown into a pond the wave motion in the immediate neighbor- 
hood of the disturbance is excessively complicated, but the waves 
become a clearly defined series of ridges at a considerable distance 
from the disturbance. 

Consider a region AB, Fig. 270, on the surface of a pond, C 
being a point at which a pebble is dropped into the pond. The 



C 




Fig. 270. 

above-mentioned fact that the waves from C resolve themselves 
into clearly defined ridges at a distance from C means that all 
points of the water surface which lie in a certain line AB rise and 
fall together, or, in other words, the line AB on the water surface 
moves up and down as a whole as the waves pass hy. Such a line 
is called a wave front. 

Sound waves in the air and light waves in the ether always re- 
solve themselves at a great distance from the disturbance into a 
clearly defined layer or series of layers {if they could hut he seen), 
and an indefinitely thin portion of such a layer moves up and down 
or to and fro as a whole, and is called a wave front. 

The direction of progression of a water wave is at right angles 
to its front. The direction of progression of a sound wave or 
light wave is at right angles to its front.* 

A wave which has a plane wave front is called a plane wave. 

* When the medium through which the wave passes has different properties in 
different directions, the direction of progression may not be at right angles to the 
front. Thus, in a substance Hke wood which has a grained structure, a sound 
wave does not in general progress in a direction at right angles to its front, in a 
crystal like Iceland spar a light wave does not in general progress in a direction 
at right angles to the front. 






SOME USEFUL IDEAS OF WAVE MOTION. 



355 



C 



A wave which has a spherical wave front is called a spherical 
wave, 

250. Huygens* principle. — Let AB, Fig. 271, be the instan- 
taneous position of a wave which has coxiie from a disturbance at 
C. Thedisturbance which is produced later at the point p when 
the wave reaches that point is of 
course to be thought of as having 
come originally from C; it may he, 
however, considered to have come from 
the disturbance which constitutes the 
wave AB. In this case, each point 
of the wave AB is to be considered 
as a center of disturbance from 
which a spherical wave emanates. 
The waves which thus emanate from 

each point of a primary wave are called secondary waves or 
wavelets. The actual disturbance produced at p is the result- 
ant of the effects of all these wavelets. 

251. Huygens' construction for wave front. — Let AB, Fig. 
272, be the front of a wave which is advancing towards A'B\ 
and let it be required to find the wave front after a time has 
elapsed during which the wave has traveled a distance r. De- 
scribe circles (spheres) of radius r about each point of the wave 





1- 
O 




356 THE THEORY OF LIGHT. 

front AB. The envelope A'B^ of these circles (spheres) is the 
required wave front. These spheres are the secondary wavelets 
described in the previous article.* 

252. The ray of light. — Consider a succession of waves WW, 
Fig. 273, traveling out from a center of disturbance 0. These 
waves progress at each point in a direction at right angles to the 
wave front, and the lines rrr along which the wave disturbance 
travels are called rays. 

A bundle of rays drawn from the various points of and at 
right angles to a small portion of a wave front is called a pencil 
of rays. 

The conception of the ray carries with it the idea that a wave 
disturbance is propagated in straight lines, that a wave disturb- 
ance will not, for example, bend round an obstacle into the 
region behind the obstacle. Water waves do, however, bend 
round an obstacle and so also do sound waves inasmuch as it is 
a common experience that a sound can be heard around a corner. 
The familiar phenomena of shadowsf and the fact one cannot 
see around a corner seem to indicate that light travels in straight 
lines. As a matter of fact, however, light bends round a corner 
in the same way that sound does but to an extent that is, under 
ordinary conditions, scarcely noticeable. This bending of light 
and sound around a corner is called diffraction. 

253. Homocentric pencil of rays. — When a portion of a wave 
front is a sector of a spherical surface (the entire wave front may 
not be a sphere) the rays from the portion intersect at the center 
of the sphere, the pencil of rays is said to be stigmatic or homo- 
centric, and the center of the sphere is called the focal point of the 
pencil. The circle WW, Fig. 274, represents a spherical wave 
emanating from a center of disturbance 0. In this case the 

* A full discussion of Huygens' principle and of Huy gens' construction for a 
wave front, together with Fresnel's improvement thereon, is to be found in Chapter 
III of Drude's Theory of Optics, translated by Mann and Millikan (New York, 
1902). 

t The phenomena of shadows are briefly discussed on pages 3 to 5 of Edser's 
Light for Students, Macmillan and Co., 1902. 



SOME USEFUL IDEAS OF WAVE MOTION. 



357 



wave Is shown as traveling towards Its convex side and the pencil 
of rays rrr Is said to be divergent. In Fig. 275 WW repre- 
sents a spherical wave which has come from a center of disturb- 
ance and has passed through a lens LL. In this case the 




^ 




point 



wave WW Is shown as traveling towards Its concave side and 
the pencil of rays rrr Is said to be convergent. 

254. Astigmatic pencil of rays. — Imagine the circle BB, Fig. 
276a, to be rotated about the axis EF so that the circle may de- 
scribe a ring-shaped surface. Consider a small portion AA of 

A 




sectional view 

Fig. 276a. 



side view 

Fig. 2'j6b. 



the surface of this ring. If normals (lines) are drawn from 
every point of the portion A A these lines will Intersect first 
along a short line at C (perpendicular to the plane of the paper 



358 



THE THEORY OF LIGHT. 



in Fig. 276a) and second along the line DD. The portion AA 
of the ring surface is sharply curved in one plane (the plane of 
the paper in Fig. 276a) and less sharply curved in another plane 
at right angles to the first (the plane of the paper in Fig. 276^). 
When a sinall portion of a wave front is shaped like the por- 
tion AA of the ring surface in Fig. 276a, the pencil of rays which 
corresponds to the portion of the wave front passes first through 
the line C and then through the line DD. Such a pencil of rays 
is called an astigmatic pencil, and the lines C and DD are called 
the focal lines oi the penaX. 

Example. — Light emanates from a point 0, Fig. 277, and 
passes obliquely through a lens LL. After passing through the 
lens the wave fronts are shaped like a portion of the surface of a 

ring (approximately), and 
the rays are focused first 
along the line C (perpen- 
dicular to the plane of the 
paper) and then along the 
line DD. If the light from 
the lens LL, Fig. 277, is 
allowed to fall on a screen 
SS a short bright line 
will appear on the screen 
if che screen passes through C or through DD. If the screen 
is between C and DD as shown in Fig. 277, a blurred spot 
will show on the screen. The light from passing obliquely 
through LL cannot be sharply focused at a point on the screen. 
The nearest approach to a sharp point is obtained when the 
screen is midway between C and DD, in which case the blurred 
spot is a small circle which is called the circle of least confusion. 



°^: 




CHAPTER XIX. 

REGULAR REFLECTION AND REFRACTION. 

255. Velocity of light in different media. Index of refraction. 

— The velocity of light is different in different media. Thus 
light travels about i| times as fast in air as in water (the index 
of refraction of water is about i|), and light travels about ij 
times as fast in air as in glass (the index of refraction of glass is 
about I J). In general, if light travels in a given transparent 
substance ijl times as fast as in air, the ratio /x is called the 
index of refraction"^ of the substance. 

The velocity of light in substances like water and glass depends 
on the color (wave length) of the light, that is the index of refrac- 
tion of a substance is different for different colors (wave-lengths) 
of light. This matter is discussed in Chapter XXIII ; the follow- 
ing discussion refers strictly to light of a single color (single 
wave-length) because in the following discussion a substance like 
water or glass is thought of as having a definite index of refraction. 

256. Application of Huygens' principle to regular reflection 
and refraction. — ^A light wave (wave front) WW comes from a 
light source and approaches a polished glass surface AB as 
shown in Fig. 278, and YYY is the position the wave would 
have reached at a certain instant y if it had not encountered 
the glass. As a matter of fact, however, the single wave WW 
breaks up into two waves one of which is thrown backwards or 
reflected by the glass surface, and the other enters the glass 
and is said to be refracted. It is required to find the position of 
the reflected wave and the position of the refracted wave at the 
instant y. 

Consider the state of affairs at an instant x, a little earlier 

. , velocity of light in a vacuum . „ , , , , , . . , 

* The ratio ; is called the absolute index 

velocity of light in a given substance 

of refraction of the substance. 

359 



36o 



THE THEORY OF LIGHT. 



than the instant y. At this instant x the wave passes through 
the point p on the glass surface, the point p is therefore a center 
of disturbance at the instant x, and a Huygens' wavelet starts 





Fig. 278. 



Fig. 279. 



out from p at this instant. At the later instant y, the wavelet 
from p has had time to travel the distance pg in air [or the 

distance - {qp) in glass]. Therefore (a) the portion of the wave- 

let from p which is in air is a hemisphere of which the radius is 
equal to pg^, and this hemisphere is represented by the semi- 
circle aaa in Fig. 279, and {h) the portion of the wavelet from 
p which is in glass is a hemisphere of which the radius is equal to 

- X {pg), and this hemisphere is represented by the semi-circle 



in Fig. 279. 

Therefore if we draw a system of circles (spheres) tangent to 
YYY and with their centers on AB, the envelope of these circles 
(spheres) will be the reflected wave as shown in Fig. 280 ; and if we 
replace each of the circles in Fig. 280 by a circle of which the 
radius Is i/^t as great we will have the system of circles (spheres) 
in Fig. 281 of which the envelope Is the refracted wave. 

In all that follows the Huygens wavelets will be spoken of 
simply as circles as shown in the diagrams. 



REGULAR REFLECTION AND REFRACTION. 



361 



257. Reflection of a plane wave from a plane surface. — Let 

WW in Fig. 282 represent a plane wave approaching a plane 
polished glass or metal surface AB, and let YY be the position 



reflected wave 




reflection 

Fig. 280. 




refracted wave^ 



refraction 

Fig. 281. 



this wave would have reached at a given instant y if it had not 
encountered AB. Then, according to Art. 256, aa is the 
position of the reflected wave at the instant y. The incident 
rays are perpendicular to WW, or YY, and the reflected rays 
are perpendicular to aa; and it is evident from the figure that 



incident ray 



jT^reflected ray 




reflecting surface 



Fig. 282. 



the angle i (called the angle of incidence) is equal to the angle r 
(called the angle of reflection) . 

258. Reflection of a spherical wave from a plane surface. — Let 

0, Fig. 283, be a luminous point from which spherical waves 



362 



THE THEORY OF LIGHT. 



emanate, let AB be a plane reflecting surface, and let FF be 
the position which would be reached at a given instant, y, by a 
spherical wave from were it not for the reflecting surface AB. 
It is required to find the position of the reflected wave at the 

given instant y. About 
each point of the surface 
AB describe a circle tan- 
gent to F F. The envel- 
ope aa of these circles is 
the reflected wave. 

As is evident from the 
symmetry of the figure, 
the reflected wave aa is 
a sphere exactly like YYj 
and the center of curva- 
ture of the reflected wave 
is at the point 0'. There- 
fore, when light from a lu- 
minous point is reflected from a plane surface the reflected 
light appears to come from a fictitious luminous point 0', This 
point 0' is called the image of O. The line 00' is perpen- 
dicular to the plane of the reflecting surface and bisected thereby. 





"'.0 



m»mh}> \ Hiiimi»)r»)i»})»»»ujij»»uM'»}> ifmf>r'mmp^ 






Fig. 284. 



Fig. 285. 



REGULAR REFLECTION AND REFRACTION. 363 

Figure 284 shows the reflection of light from a small plane 
mirror AB. The relativ^e positions of and 0' in Fig. 284 are 
the same as if the mirror were large. 

259. Image of an object in a plane mirror. — An ordinary object 
Is a group of luminous points in so far as its light-giving properties 
are concerned. Consider such an object 0, Fig. 285. The 
light from the various points of this object, after reflection from 
a plane surface AB, or from a portion of such a surface, appears 
to come from a similar group of points 0\ This group of 
fictitious luminous points 0' is called the image of the object 0. 

260. Three simple cases of reflection from curved surfaces. 

Case I. The spherical mirror of small aperture. — Figure 286 
shows a mirror MM the reflecting surface of which forms part 




^=^=^'^ 4^'Il'_«yc's of mirror 

Center of 



curvature 
of mirror 



Fig. 286. 



of a sphere SS with Its center at A. The diameter of MM 
is small in comparison with the radius of the sphere SS. Such a 
mirror Is called a spherical mirror of small aperture, and the light 
from a luininous point a is very nearly concentrated at the 
point h. The theory of spherical mirrors of small aperture is 
very similar to the theory of lenses, and inasmuch as the theory 
of simple lenses Is quite fully discussed in Chapter XX, it is not 
worth while to discuss spherical mirrors. 

Case II. The parabolic reflector. — The heavy line In Fig. 287 
represents a mirror of which the reflecting surface Is a paraboloid 
of revolution. A beam of parallel rays (parallel to the axis of 
the paraboloid) is concentrated at the focus of the paraboloid ; or 



364 



THE THEORY OF LIGHT. 



light emanating from the focus is reflected by the paraboloid as 
a beam of parallel rays (parallel to the axis of the paraboloid). 




Fig. 287. 



Fig. 288. 



center of curvature 
of mirror 

^ \^ 

F A 



The parabolic reflector is extensively used in locomotive head- 
lights and in searchlights. 
Case III. The ellipsoidal 
reflector. — The heavy line 
in Fig. 288 represents a 
hollow metal shell the in- 
terior (polished) surface of 
which is an ellipsoid of 
revolution (prolate ellip- 
soid). Light emanating 
from one focal point of 
the ellipsoid is concen- 
trated at the other focal 
point. Vaulted ceilings 
sometimes approximate 
very closely to the ellip- 
soidal form, and a very 
faint sound at one focal 

point can be heard distinctly at the other focal point. 

261. General case of the reflection of a plane wave from a 
spherical surface. — Let MM, Fig. 289, represent a polished 




Fig. 289. 



REGULAR REFLECTION AND REFRACTION. 



365 



spherical surface. A plane wave comes from the right, and YY 
is the position which this wave would have reached at a certain 
instant y if it had not encountered the mirror MM. Using the 
method of Art. 256 we find the position aaa of the reflected wave 
at the instant y. The reflected wave aaa, Fig. 289, is not 
spherical. Indeed Figure 290 is a more extended drawing which 




Fig. 290. 

shows the reflected wave aa with two cusps, and the heavy lines 
in Fig. 291 show two successive positions of the reflected wave. 
The small arrows in Figs. 289, 290 and 291 show the direction of 
travel of the reflected wave. The cusps of the reflected wave 
travel along the dotted lines CC in Fig. 291, and these dotted 
lines CC* are called the caustics of the reflected wave. 

* These lines, of course, represent a surface, for Fig. 291 represents a 3-dimensional 
arrangement. Rotate the figure about the axis of the mirror and the surface 
generated by the lines CC in Fig. 291 is the caustic surface. 



366 



THE THEORY OF LIGHT. 



F 



axis of _ 
mirror 



An important property of the caustic curve is shown in Fig. 
292. The center of curvature of the small arc A A lies on the 
caustic surface at the point c. Indeed the pencil of rays corre- 
sponding to the small portion 
of the wave front A A is an 
astigmatic pencil. One focal line 
of the pencil is at c (perpen- 
dicular to the plane of the paper) 
and the other focal line is dd. 

4 262. Refraction of a plane wave 

at a plane surface. — Consider 
a plane wave WW, Fig. 293, 
approaching the plane surface pO 
of a piece of glass. At a cer- 
tain instant y this wave would 
have reached the position YY if 
it had not encountered the glass. 
About each point p on the sur- 
face of the glass describe a circle of which the radius is djix, 
where d is the distance ps and /x is the index of refraction of 
the glass. Then the envelope of these circles represents the 
required refracted wave as explained in Art. 256. 




Fig. 291. 




center of curvature 
of mirror 



axis of mirror 



Fig. 292. 

The incident rays are at right angles to WW and to YY 
and the refracted rays are at right angles to aa. 



REGULAR REFLECTION AND REFRACTION. 



367 



Snell's law. — The angle i between an incident ray and the 

normal to the refracting surface is called the angle of incidence, 

and the angle r between a refracted ray and the normal to the 

refracting surface is called the angle of refraction, and, as was 

first shown by Snell, 

sin ^ 

-. = M (97) 

sm r 



where /* is the index of refraction of the glass. To establish 
equation (97) let us consider Fig. 294 which shows the more 




y ^':^fk 




-refracted ray i^'f"'- sj 
Fig. 293. Fig. 294. 

important parts of Fig. 293. The right triangles pOs and pOs* 
have a common side pO, and the angles at are equal to i 
and r respectively. Therefore: 



and 



so that 



sm.=- 



smr=- 

sin i d 
sin r d' 



(i) 



(ii) 



(iii) 



but d' is written for dlix, so that equation (iii) reduces to 
equation (97). 



368 



THE THEORY OF LIGHT. 



Refraction of a pencil of parallel rays (a plane wave) by a 

prism. — Figure 295 shows the path of a ray of Hght through a 

prism of glass. The total 

angle of deflection a depends 

on the index of refraction of 

the glass, on the angle of 

the prism and on the direction 

of the ray in the prism. 

When the ray in the prism 

is equally inclined to the two 

faces of the prism as shown 

in Fig. 295 then the total deflection a is a minimum, and in 

this case: 

sin i(0 + «) . , ^.^ 

(98)* 




Fig. 295. 



sin I0 



= M 



The index of refraction of a sample of glass (for light of a given 
color or wave-length) is usually determined by making a prism 





Fig. 296. Fig. 297. 

from the sample, measuring the angle of the prism, and ob- 
serving the angle of deflection a ; whence the index of refraction 
II may be calculated from equation (98). 

263. Total reflection. — ^When a ray of light passes from glass 
into air, as shown by the ray Opq^ in Fig. 296, then, according to 
Snell's law, we have: 

sin i — II sin r (i) 

* See Preston's Theory of Light, page 123. 



REGULAR REFLECTION AND REFRACTION. 369 

where r is the angle between the normal to the surface and the 
ray which is in the glass, and i is the angle between the normal 
to the surface and the ray which is in the air. 

For example, let /x = 1.5; then sin ^' must be unity (or i 
must be 90°) when sin r = 2/3. Therefore as sin r approaches 
the value 2/3 the emergent ray pq approaches the direction pq\ 
If sin r is greater than 2/j, then, according to equation (i), sin i 
would have to be greater than unity, which is impossible. In fact 
the ray Op in the glass is completely reflected (in the direction 
ps) when ix sin r is greater than unity. This phenomenon is 
called total reflection. It is the cause of the brilliant silvery 
appearance of the surface of water in a tumbler when viewed 
obliquely from below. 

The dotted line in Fig. 297 shows the critical position of the 
ray Op in Fig. 296. When the angle r, Fig. 296, is less than the 
critical value part of the light is reflected back into the glass, 
when the angle r is greater than the critical value, the light is 
totally reflected back into the glass. 

264. Refraction of a spherical wave at a plane surface. — The 
point in Fig. 298 represents a luminous point in glass (or 
water), WW represents the position a spherical wave from 
would have reached at a given instant if it had continued traveling 
in glass (or water), and WW shows the actual position of 
the wave in the air at the given instant. The wax^e WW is 
not spherical, and a pencil of rays corresponding to the small 
portion aa of the wave front is an astigmatic pencil with one 
of its focal lines at C (perpendicular to the plane of the paper) 
on the caustic, and the other focal line is at DD on the axis of 
symmetry of the figure. A small portion of the wave front 
near V is sensibly spherical, and the pencil of rays corresponding 
to a small portion of the wave front at F is a homocentric pencil 
with its focal point at F. 

A very striking experiment Is the following: A fine bit of 
chalk is placed at the bottoin of a basin of water (at O, Fig. 298). 
With the eyes at aa the bit of chalk is seen at DD if the line 
25 



370 



THE THEORY OF LIGHT 



joining the eyes is horizontal, and it is seen at C tf the line joining 
the eyes is vertical. The familiar broken appearance of a straight 




Fig. 298. 

oar in still, clear water is due to the fact that any point, like 0, 
Fig. 298, of the submerged portion of the oar seems to be raised 
to DD. 



CHAPTER XX. 

SIMPLE LENSES. 

265. A lens is a portion of glass* bounded by plane or spherical 
surfaces as shown in the accompanying figures in which R 




^^ axis of 
lens 



Fig. 299. 



y 

y/ axis of 
lens 



Fig. 300. 




yL /R 

^^'axis of^ 

lens 




Fig. 301. 



represents the radius of curvature of the right face of the lens 
and L represents the radius of curvature of the left face of the 
lens. 




"^^ lens "^^ 7^" 

^X W L/ 



Fig. 302. 



^ axis of 

v^ lens 



L^ 



\ 



\ 







Fig. 303- 



The axis of the lens is the line joining the centers of curvature 
of the faces of the lens, or it is a line passing through the center 
of curvature of one face and cutting the o ther face at right angles 
as shown in Figs. 299 and 300. 

In the manufacture of lenses the rough lens is generally 
moulded from the desired quality of glass and then annealed. 

* Or any transparent substance. 

371 



372 



THE THEORY OF LIGHT. 



Each face of the lens is then accurately shaped by grinding the 
lens and a matrix together with every possible variety of sliding 
motion thus bringing the surface of the lens automatically to 
almost perfect spherical shape. The polishing is accomplished 
by using finer and finer grinding material, usually powdered 
emery, and ending with rouge. The matrix is usually of iron 
and in the later stages of the grinding and polishing the matrix is 
lined with a layer of stiff pitch with cross grooves cut in its 
surface.* 

Converging lens. — Figure 304 shows a train of plane light 
waves A A passing through a lens which is thick at the center 
and thin at the edge. The waves travel slower in glass than in 
air, so that the portions of the waves which pass through the 
thick part of the lens fall behind the portions which pass through 
the thin part of the lens; therefore the waves BB have concave 
fronts as shown, and they are concentrated at a certain point as 
indicated in the figure. A lens which is thick at the center and 
thin at the edge is called a converging lens. 





Fig. 304. 



Fig. 305. 



Diverging lens. — Figure 305 shows a train of plane light waves 
AA passing through a lens which is thin at the center and thick 
at the edge. In this case the central portions of the waves get 
ahead of the edge portions as they pass through the lens; there- 

* For a simple account of the processes of lens grinding see Lockyer's Stargazing, 
pages 1 1 7-138; see also "How to make a Refracting Telescope" in the Scientific 
American Supplement, Nos. 581, 582 and 583 (February 19 and 26 and March 5, 
1887), pages 9283-9285, 9296-9299 and 9312-9316. A very interesting account of 
the manufacture of a large reflecting telescope is given by Henry Draper in the 
Smithsonian Contributions to Knowledge, No. 180, Vol. XIV, 1865. 



SIMPLE LENSES. 



373 



fore the waves BB, Fig. 305, have convex fronts as shown, and 
they appear to have come from a certain point as indicated by 
the dotted lines in the figure. A lens which is thin at the cefiter 
and thick at the edge is called a diverging lens. 

It is best in lens diagrams to show the rays without showing the 
wave fronts. Thus Figs. 306 and 307 are duplicates of Figs. 304 
and 305 respectively, but showing rays instead of waves. 




IX 


f 




A ^ 


i * ' ,^-' 


i ,-''■' 


^-^ " 


f"--^^ 






~~- -~_ 




Fig. 307. 

266. Focal points and focal lengths. — ^The point F in Fig. 306 
at which the pencil of parallel rays AA is focused by the lens is 
called the jocal point of the lens, and the distance / is called the 
jocal length of the lens. 

The point F in Fig. 307 from which the pencil of parallel rays 
A A appears to have come after passing through the lens is 
called the jocal point of the lens, and the distance / is called 
the jocal length of the lens. 

A lens has, of course, two focal points, one on each side of the 
lens, and the focal length / is the distance from the lens to either 
focal point. 

The action of a converging lens as shown in Fig. 306 is evidently 
very different physically from the action of a diverging lens as 
shown in Fig. 307, and yet the term focal point is applied to both, 
and the term focal length is applied to both. In all lens equations 
the focal length of a converging lens appears as a positive quantity 
and the focal length of a diverging lens appears as a negative 
quantity. 

267. Limitations of this chapter. The ideal simple lens. — 
Everyone is familiar with ordinary moving pictures in which 



374 



THE THEORY OF LIGHT. 



images of a rapid succession of small transparent photographs 
are projected on a screen by a lens. This formation of an image 
by a lens is also exemplified in the photographic camera; an 
image of the object to be photographed is projected by the 
camera lens on sensitive plate and this image is made permanent 
as a photograph. The projecting lens of a moving picture 
machine and photographic camera lenses of high grade are not 
simple lenses, they are coinpound lenses as explained in Chapter 
XXII. A simple lens produces a sharply defined image of an object 
only when the lens is very thin in comparison with its diameter,"^ 
when the light passes through the lens in a direction nearly parallel 
to the axis of the lens, and when light of one color {one wave-length) 
is used, and the discussion of simple lenses in this chapter is 
subject to these limitations. In the diagrams, however, the 
lenses are shown as having considerable thickness, and the rays 
of light are frequently shown as being greatly inclined to the 
axis, all for the sake of clearness. 

268. Conjugate points. — The image on a moving picture screen 
is called a real image, and in the formation of a real image by a 
lens all of the light that reaches the lens from any point a of 
the object is focused by the lens at the corresponding point h 
of the image, as indicated in Fig. 308, and the two points a 
and h constitute what is called a pair of conjugate points with 



axis of lens y'' /A. 





Fig. 308. 



Fig. 309. 



respect to the lens. Indeed a pair of conjugate points is shown 
in each of the accompanying figures. 

* In order that this statement may be thus made the lens is supposed to come 
to a sharp edge as in Figs. 299, 301 and 302, or to have zero thickness at the center 
in Figs. 300 and 303. 



SIMPLE LENSES. 



375 



In Fig. 308 both points are said to be real; light from a is 
focused at h, or Hght from h is focused at a by the lens. 

In Fig. 309 a is real and h is said to be virtual; light from 
a appears to have come from h after passing through the lens, 
or the beam BB trending towards b (but never reaching it) is 
focused at a by the lens. 

In Fig. 310 a is real and h is virtual. 

In Fig. 311 a and h are both virtual ; the beam A A trending 
towards a (but never reaching it) appears to have come from h 




Fig. 310. 



Fig. 311. 



after passing through the lens, or the beam BB trending towards 
h (but never reaching it) appears to have come from a after 
passing through the lens. 

Remark i. — The focal pomt of a converging lens, as shown in 
Fig. 306, is real; the focal point of a diverging lens, as shown in 
Fig. 307, is virtual. 

Remark 2. — Every ray of an ^-pencil in Figs. 308, 309, 310 
and 311 passes through point a, and every ray of a 5-pencil 
passes through point 6, the ray being extended straight through 
the lens as a dotted line in case of a virtual point. 

269. The geometry of image formation. — ^The complete geom- 
etry of image formation by a simple lens can be developed from 
the facts which are set forth in Figs. 306 and 307 together with 
the following proposition : 

The straight line connecting any pair of conjugate points passes 
through the center of a simple lens. Consider the particular ray 
R, Fig. 312, which comes from any given point a (or which goes 
towards a if a is a virtual point) and passes through the center of 
the lens. This ray cuts the two surfaces of the lens at points where 



376 



THE THEORY OF LIGHT. 



the surfaces are parallel to each other.* Therefore the lens acts 
like a flat glass plate on the incident ray R as shown in Fig. 313, 





Fig. 312. 



Fig. 313. 



and consequently the emergent ray R' is parallel to R, but 
displaced slightly sidewise. The sidewise displacement is, how- 
ever, negligible because the lens is supposed to be indefinitely 
thin; therefore RR^ is sensibly one straight line. But R 
comes from (or goes towards) a, and R' passes through (or 
appears to come from) b which is the conjugate of a. Therefore 
the points a, C and b lie on the straight line RR\ 

Geometrical constructions. — To understand Figs. 314 and 315 
it must be remembered that any incident ray parallel to the 
axis of a converging lens passes, on emergence, through the focal 
point of the lens. 

X X. 




axis of 



lens 




lens 



Fig. 314. 



Fig. 315- 



Given a real point a to find Given the virtual point a to 

its conjugate b. Draw ray aX find its conjugate b. Draw 

parallel to the axis of the lens; the ray P parallel to axis of 

draw XF; then draw the ray lens as if it were coming to 

aC; and b is at the intersec- point a (which it never 

tion of XF and R. reaches) ; draw the ray XF; 

* If the surfaces of the lens are of unequal curvature there is a point C between 
them such that a line through C cuts the two surfaces at points where they are 
parallel to each other. 



SIMPLE LENSES. 



377 



then draw the ray R straight 
through C to a; and b is at 
the intersection of XF and R, 
The point b thus located is 
always real, because both 
points of a conjugate pair can- 
not be virtual in the case of a 
converging lens. 



The point b thus located is 
real in Fig. 314, but it may be 
real or virtual; if the point b is 
virtual the ray XF must be ex- 
tended to the left (as a dotted 
line) and the point of intersec- 
tion of XF and R will be on 
the same side of the lens as the 
point a. 



To understand Figs. 316 and 317 it must be remembered that 
any incident ray parallel to the axis of a diverging lens appears, 
on emergence, to have come from the focal point back of the lens. 




axis of ■" 
lens p 



Fig. 316. 

Given a real point a to find 
its conjugate b. Draw aX 
parallel to axis of lens; draw 
the ray FX as if it had come 
from F; then draw aC; and 
b is at the intersection of FX 
and R, 



The point b thus located Is 
always virtual, because both 
points of a conjugate pair can- 




axis qf_ __^ 

lens -^ ~ 



Fig. 317- 

Given a virtual point a to 
find its conjugate b. Draw 
the ray P parallel to axis of 
lens as if it were coming to 
point a (which it never 
reaches); draw the ray FX 
as if it had come from F; then 
draw aC; and b is at the 
intersection of FX and R. 

The point b thus located is 
virtual in Fig. 317, but it may 
be real or virtual. If it is 



378 



THE THEORY OF LIGHT. 



not be real in case of a diverg- real, it will be on the same 
ing lens. side of the lens as the virtual 

point a in Fig. 317. 




H-J -^ -H 



Fig. 318. 




->i 



Fig. 319. 



270. Relative dimensions of object and image. — Figures 318 
to 321 show object and image for the four cases shown in Figs. 





Fig. 321. 

308 to 311 respectively. The diameter of the object is to the 
diameter of its image as the distance a is to the distance b. 

271. The focal length of a converging lens is given by the 
equation 

where / is the focal length of the lens, d is the diameter of the 

lens measured to the sharp 
edge of the lens, h is the 
thickness of the lens at the 
center, and ju is the index of 
refraction of the glass. 

Proof. — Consider a plane 
wave A A, Fig. 322. While 
the central portion of this 
wave goes through thickness 

* It is not worth while considering the very slight modification of this equation 
for the case of a diverging lens. 




'this distance very small 

Fig. 322. 



SIMPLE LENSES. 379 

h of glass, the portions o; the wave which go through the ex- 
treme edge of the lens will go ix times as far in air. Therefore 
the effect of the lens on a wave is to retard the central part of the 
wave causing it to fall a distance {fih — h), or (fx — i)h, behind 
the edge portion ee of the transmitted wave. 

The transmitted wave ee is sensibly spherical, the chord ee 
is sensibly equal to the diameter d of the lens and the versed 
sine is (/i — i)h. Therefore from the triangle eFc we have 

i' = (^)' + [/-U-iW* 

or, expanding, we have 

d^ 
/^ = — + /2 — 2f{n — i)h + a term containing h^ (i) 

4 

But h is very small in comparison with d and / so that the 
term containing h^ may be neglected, so that solving equation 
(i) for / we get equation (99). 

272. Proposition. — Let / be the focal length of a converging 
lens, and a and b the respective distances from the lens to a 
pair of conjugate points. Then we have 

ra + b ^'"^^ 

This equation can be derived from the geometrical relations in 
Fig. 314 but it is more instructive to derive it as follows: 

Consider a spherical wave WW, Fig. 323, which has reached the 
lens from the point A which is at the distance a from the lens. 
Consider a portion of this wave of which the diameter is d 
(the same as the diameter of the lens) and let k be the distance 
shown in the figure. Then 

d^ 

* The distance eF is here taken to be equal to /, and the distance cF is taken 
to be equal to / — (ju — i)^. 



380 



THE THEORY OF LIGHT. 



as explained in the derivation of equation (i) of the preWous 
article. 




— this distance is assumed to be 
negligibly small 

Fig. 323. 

In passing through the lens the central part of the portion WW 
of the incident wave falls behind the edge portions by the amount 
(ju — i)h, as explained in the previous article, and therefore the 
central portion of the transmitted wave W'W' is a distance 
(/i — i)h — k behind the edge portions. The transmitted wave 
W'W' is sensibly spherical and it is concentrated at its center 
of curvature B which is at a distance h from the lens {h is 
also the radius of curvature of W'W'). Therefore, by the 
argument which gave equation (i) of Art. 271, we have 

^2 



h = 



8[(m - i)h - k] 



(il) 



Substituting the value of / from equation (99) and the v^alues of 
a and b from equations (i) and (ii) in equation (100) we get 

the identity 

8(m - i)h Sk . 8[(m - i)h - k] 



= 1^ + 



so that equation (100) is established. 

Note. — ^The above derivation applies explicitly to a converging 
lens, but equation (100) applies to any simple lens provided / 
is considered as negative in case of a diverging lens. In the 
use of equation (100) the distance from the lens to a virtual 
point must be considered as negative. 



CHAPTER XXI. 



SIMPLE OPTICAL INSTRUMENTS. 

273. Simple optical instruments. — In order to understand 
simple optical instruments the various component lenses of such 
instruments should be thought of as ideal simple lenses, although 
compound lenses are always used. Thus the projecting lens of 
a moving picture machine and the lens of a photographic camera 
are usually compound lenses as stated in Art. 267. 

The essential optical features of the projection lantern (the 
moving picture machine) are shown in Fig. 324. The light from 



R i 




^^f- ^. 



\^ 



Fig. 324. 

a lamp L passes through the two lenses CC, through a trans- 
parent picture SS and through a lens which projects an 
image of 55 upon a wall or screen. The lenses CC are called 
condensing lenses, the transparent picture SS is called a lantern 
slide, and the lens is called the objective or object lens of the 
lantern. 

274. The eye.* — Figure 325 shows a horizontal section of the 
human eye. The tough outer coating of the eye-ball is sharply 

* A good discussion of the structure of the eye and its functions; of the ophthal- 
moscope; of binocular vision and the stereoscope; and of the persistence of vision 
and the stroboscope is given in Edser's Light for Students, pages 159—196, Macmillan 
and Company. See also Miiller-Pouillet's Lehrbuch der Physik, Vol. II, part I 
(on light by Otto Lummer), pages 580-643. The great work on this subject is 
Helmholtz's Handbuch der physiologischen Optik, Leipzig, 1896. 

381 



382 



THE THEORY OF LIGHT. 



curved and transparent in front forming the cornea NN, Be- 
tween the cornea and the crystalline lens A is a clear watery- 
fluid, the aqueous humor B. Behind the crystalline lens and 
filling the remainder of the eye-ball is a clear semi-fluid substance, 
the vitreous humor C. The front surface of the cornea and the 
two surfaces of the crystalline lens are sensibly spherical and 

they constitute a compound 
lens which projects an image 
of external objects upon a sen- 
sitive membrane, the retina I. 
The retina consists of a great 
number of minute end-organs 
of nerve fibers which enter the 
eye in a bundle at 0, consti- 
tuting what is called the optic 
nerve. The most sensitive part 
of the retina is a slightly de- 
pressed place at p which is 
called from its color the yellow 
spot. In the yellow spot the 
terminal organs of the optic nerve are packed very close together, 
and in order to see an object distinctly its image must fall upon 
this spot. Light which enters the eye passes through an aperture 
bb in a muscular membrane which is called the iris. The aper- 
ture bb is called the pupil of the eye. 

Accommodation. — In the photographic camera the distance of 
the lens from the sensitive plate is adjustable so that distant ob- 
jects or near objects may be sharply focused upon the sensitive 
plate at will. In the eye, however, the distance from the crystal- 
line lens to the retina is invariable and the lens of the eye is 
focused by the action of a muscle M which surrounds the crystal- 
line lens like a barrel hoop. The lens is, in youth, a bag of 
semi-fluid jelly, and when looking at a near object the muscle 
•M squeezes the lens and makes it thicker at the center. This 
adjustment of the focus of the lens of the eye to give distinct 




Fig. 325. 



SIMPLE OPTICAL INSTRUMENTS. 383 

vision is called accommodation. Ordinarily the eye has power of 
accommodation for objects at any distance greater than 15 centi- 
meters from the eye. The distance of most distinct vision for 
the normal eye is about 25 centimeters. The accommodation of 
the eye is sensibly invariable for all distances exceeding eighty or 
ninety meters. 

Imperfections of the eye. — Some persons can accommodate the 
eye to distant objects with great effort or not at all; some persons 
can accommodate the eye to near objects with great difficulty 
or not at all; some persons can accommodate the eye so as to 
see vertical lines or horizontal lines sharply but not so as to see 
both vertical and horizontal lines distinctly at the same time. 
A person who cannot accommodate his eyes to distant objects 
but who can accommodate his eyes to near objects is said to be 
near sighted. A person who can accommodate his eyes to dis- 
tant objects but who cannot accommodate his eyes to near objects 
is said to be far sighted. Near-sightedness is relieved by the 
use of spectacles with diverging lenses, far-sightedness is relieved 
by the use of spectacles with converging lenses. Inability to see 
vertical lines and horizontal lines simultaneously is due to inac- 
curate "centering"* of the eye lenses or to more or less deviation 
from true spherical shape of the various refracting surfaces of the 
eye, and it is called astigmatism. Astigmatism is corrected by 
the use of spectacles having cylindrical surfaces. 

The diopter. It has become the universal practice among 
optometrists to express the "power" of a spectacle lens by giving 
the reciprocal of its focal length in meters, and a lens of one 
meter focal length Is said to have a power of one diopter. Thus 
a lens of two meters focal length has a power of half a diopter, 
a lens of half a meter focal length has a power of two diopters. 

275. Apparent size of objects. Visual angle. — ^An object ap- 
pears large when its image covers a large portion of the retina. 
Lines drawn from the extremities of the object through the 

* A compound lens system is said to be centered when the centers of curvature 
of all the spherical surfaces lie on the same straight line. 



384 



THE THEORY OF LIGHT. 



center* of the eye lenses, as shown in Fig. 326, pass through the 
extremities of the image on the retina. The angle a between 
these lines determines, therefore, the size of the image on the 
retina; this angle is called the visual angle of the object and it 
is taken as the measure of the apparent size of the object. 




Fig. 326. 

It must be remembered that the angle a in Fig. 326 is always 
quite small because an image of an object must fall upon a very 
small portion of the retina (the yellow spot) if one is to see the 
whole object distinctly. In the diagrams of the eye, however, 
Figs. 326 and 327, the angle a is shown rather large for the sake 
of clearness. 

276. The simple microscope. Definition of magnifying power. 

— The simple microscope or magnifying glass is a converging lens 
which is held as near to the eye as possible, the object to be 
examined being moved up until it is seen most distinctly. The 
eye is then looking at an enlarged virtual image of the object and 
this virtual image is at the distance of most distinct vision from 
the eye, say, 25 centimeters, as shown in Fig. 327. The light 
from the point a of the object appears to have come from the 
point h after passing through the magnifying glass. 

The magnifying power of a microscope is defined as the ratio 
of the apparent size {visual angle) of an object as seen with the 
microscope to its apparent size {visual angle) as seen with the 
naked eye at a distance of 25 centimeters. 

* This statement is made as if the compound lens effect of cornea and crystalHne 
lens were equivalent to an ideal simple lens, which is not strictly true. 



SIMPLE OPTICAL INSTRUMENTS. 385 

Proposition. — The magnifying power of a magnifying glass is 



W = — r + I 



(lOl) 



in which / is the focal length of the magnifying glass in centi- 
meters. The eye is assumed to be accommodated for a distance 
of 25 centimeters, that is to say, the plane b¥ in Fig. 327 is 
assumed to be 25 centimeters from the eye. 

Proof of equation (loi). — The angle subtended by the object 
O at a distance of 25 centimeters is less than the angle aCa' in 
the ratio of 25 centimeters to a, all angles being very small. 
Therefore the angle aCa' is 2^1 a times the visual angle the object 
would have at a distance of 25 centimeters from the naked eye. 



eye 




Fig. 327. 

The visual angle of the object as seen through the magnifying 
glass is the angle between lines drawn from the points b and &' 
to the center of the lens of the eye, but the distance between 
the magnifying glass and the eye is quite small, and if we 
neglect this distance the visual angle of the object as seen through the 
magnifying glass may be considered to be equal to bCb' or aCa'. 

But, as stated above, the angle aCa' is 25/a times as great 

as the visual angle the object would have as seen by the naked 

eye at a distance of 25 centimeters. Therefore the magnifying 

power of the magnifying glass is 25/a. Now bb' is a virtual image 

26 



386 THE THEORY OF LIGHT. 

so that using — 25 centimeters for h in equation (100) we have 



whence 



- = - - — (i) 

f a 2S 

25 25 

W = — = -r- + I 
O' J 



Note. — When the eye is accommodated for parallel rays the 
distance a in Fig. 327 is equal to /, and the magnifying power 
of the magnifying glass becomes 25//. 

Note. — A high-power magnifying glass must have a very short 
focal length according to equation (loi), and it must be held 
very near to the object to be examined and very near to the eye. 
Magnifying glasses are inconvenient and unsatisfactory with 
magnifying powers exceeding about 25 or 30 diameters (focal 
lengths less than about i centimeter). Therefore for higher 
powers the compound microscope is used. 

277. The compound microscope* consists of a lens A, Fig. 
J28, which forms an enlarged real image I of an object 0, and a 



I 

..V 






I — -^- 



a. 



k 



Fig. 328. 



magnifying glass B for viewing this image. The lens A is 
called the object-glass of the microscope and the lens B is called 

* See "The Microscope and Its Revelations," by W. Carpenter (revised by W. 
H. Dallinger), London, 1901. An interesting outline of the historical development 
of the microscope is given in Hastings' Light, pages 83-110, Charles Scribner's Sons- 
190 1. Very good practical directions for the use and care of the microscope are pub- 
lished by the Bausch and Lomb Optical Company, of Rochester, New York. This 
publication is in the form of a small pamphlet and it consists of extracts from a larger 
work "The Manipulation of the Microscope," by Edward Bausch. A very com- 
plete discussion of the microscope by Czapski is given in Winkelmann's Handbuch 
der Physik, Vol. VI, pages 328-373. 



SIMPLE OPTICAL INSTRUMENTS. 387 

the eye-piece. In modern high-grade microscopes the object-glass 
and eye-piece both consist of compound lenses. No attempt is 
made in Fig. 328 to show the actual paths of the rays of light 
through the microscope; the dotted lines are drawn from the 
extremities of the object through the center of the object-glass, 
and these lines therefore determine the extremities of the image. 
The image I is viewed through the magnifying glass (eye-piece) 
B exactly as if it were an ordinar}^ object like in Fig. 327. 
Proposition. — ^The magnifying power of a compound micro- 
scope is 



=K?-) 



in which a and h are the respective distances of object and 
image from the center of the object-glass as shown in Fig. 328, 
and / is the focal length of the eye-piece. 

Proof of equation {102). —The image / in Fig. 328 is b/a times 
as large as the object, and the eye-piece (an ordinary magnifying 
glass) makes this image appear (25// -\- i) times as large as it 
would appear to the naked eye at a distance of 25 centimeters, or 
b/a{2s/f +1) times as large as the object itself would appear 
at a distance of 25 centimeters from the naked eye. 

278. The telescope* consists of a long-focal-length lens 0, 
which forms a real image i of a distant object, and a magnifying 
glass for viewing the image, a.s shown in Fig, S29. The lens is 
called the object glass of the telescope and the lens E is called 
the eye-piece. In modern high-grade telescopes the object- 
glass and the eye-piece are both compound lenses. The dotted 
lines in Fig. 329 are not intended to represent the paths of the 
rays through the telescope; but to indicate the visual angle jS 
the distant object would have if seen by the naked eye, and the 
visual angle a of the distant object as actually seen through the 

* A very interesting discussion of the telescope is given in Lockyer's Stargazing, 
pages 138-172, and in Hastings' Light, pages 53-82. A very complete discussion 
of the telescope is given by Czapski in Winkelmann's Handbuch der theoretischen 
Physik, Vol. VI, pages 386-432. 



388 



THE THEORY OF LIGHT. 



telescope. The ratio aj^ is the magnifying power of the tele- 
scope and, since the two angles a and /3 are understood to be 






/3 



—Jk'., 

ice 



A 



■n. 




f 

Fig. 329. 



y<: — ^—4 



very small, it is evident that a/jS = ///'. Therefore we have 






(103) 



in which / is the focal length of the object-glass (the object is so 
far away that the image i is very nearly at the focal point of 0), 
and f is the focal length of the eye-piece (the observer's eye 
being accommodated for parallel rays as explained in the note 
in Art. 276. 

279. The erecting telescope or spy-glass. — ^The simple tele- 
scope, the essential features of which are shown in Fig. 329, 
shows objects inverted. The spy-glass is a telescope so modified 
as to make distant objects appear erect. Figure 330 shows the 



E 



:t-¥t4 



Fig. 330. 

essential parts of a spy-glass. The object-glass forms at i 
an inverted image of a distant object, and the lens S forms at 
i' an inverted image of i or an erect image of the distant object. 
This erect image i' is viewed by the magnifying glass E. The 
introduction of the erecting lens S lengthens the telescope very 
materially and the spy-glass is consequently very long. In prac- 



SIMPLE OPTICAL INSTRUMENTS. 



389 



tice each of the lenses 0, 5 and £, Fig. 330, is a compound lens. 

280. The opera glass is a telescope of which the eye-piece is a 

diverging lens, as shown in Figs. 331 and 332. The object-glass 




Fig. 332. 



would form at i an inverted real image of the distant object 
if it were not for the interposition of the diverging lens E, the 
eye-piece. Light from a point in the object converges towards 
the point a after passing through the object-glass 0, and the 
effect of the eye-piece is to cause the pencil of rays which is 
converging towards a to appear to have come from the point h, 
that is to say, the lens E forms an inverted enlarged virtual 
image at i' of the inverted virtual image i. The image i' is 
erect, and the eye in looking through the lens E sees the distant 
object in an erect position. The action of the lens E in Fig. 
332 is as shown in Figs. 311 and 317. The lens is always a 
compound lens; but in the cheaper grades of opera glasses the 
lens E is a simple lens. 

281. The telescope with Porro prisms. — ^The great advantage 
of the opera-glass type of telescope is its shortness. An arrange- 
ment which was devised by Porro in 1852 makes it possible to 
greatly shorten the simple telescope of Fig. 329, and Porro's 
arrangement also makes a distant object appear erect. The 
essential feature of this arrangement is a pair of total reflecting 
prisms as shown in Fig. 333. A binocular telescope, or field 
glass of this type is shown in Fig. 334. 

282. The use of the telescope for sighting. — As used in the 
surveyor's transit and level, and as used in a great variety of 



390 



THE THEORY OF LIGHT. 



astronomical and physical instruments, the telescope is used 
solely for accurate sighting. Very fine wires or spider lines 
are stretched across the focal plane pp of the object-glass as 




Fig. 333. 



Fig. 334- 



shown in Fig. 335. These cross-wires are in the same plane as 
the image of the distant object (as formed by the object-glass). 
Therefore the cross-wires and the image are both seen distinctly 
by the observer as he looks through the eye-piece, and the 



object 



eye 



glass 




piece 



Fig. 335. 



observer can move the telescope until the point where the two 
wires cross each other is accurately coincident with any desired 
point a in the image ; when this is done the line drawn from the 
crossing point of the wires through the center"^ of the object-glass 
passes through the corresponding point h of the distant object, 

* The object-glass is here assumed to be an ideal simple lens. 



SIMPLE OPTICAL INSTRUMENTS. 391 

Sighting cannot be accurately done when the two "sights" 
consist of pins or point-like projections as on a gun, because it is 
impossible to focus the eye sharply on both sights and on the 
distant object simultaneously. The most accurate sighting that 
can be done by the naked eye in this way is with an error of 
about one minute of angle (about one inch at a distance of 100 
yards). With a telescope the extreme accuracy of sighting is 
with an error of a few hundredths of a second of angle, less than 
a thousandth of an inch at a distance of a hundred yards. 



CHAPTER XXII. 



LENS IMPERFECTIONS AND THEIR CORRECTION BY 
COMPOUNDING.* 

283. Focal imperfections of a perfect lens. — In the discussion 
of the theory of lens imperfections it is important to understand 
that light is focused at a more or less blurred spot instead of a 
point because of the nature of light itself, and in spite of ideal 
perfection in the performance of a lens, 

A pebble is dropped at the center of a circular tub of water, 
a series of circular ripples pass out and are reflected by the wall 
of the tub, and WW, Fig. 336 represents a wave front on one 

of the reflected ripples as it converges 
towards the point F. The wave-energy 
of the reflected ripple is not, however, 
concentrated at the mathematical point 
F. A single ripple has some breadth 
and a series of ripples has a definite 
V wave-length, and the wave-energy in Fig. 
336 is concentrated at a spot the dimen- 
sions of which are of the same order of 
magnitude as the breadth of a single 
ripple or the same order of magnitude as the wave-length of a 
series of ripples. So it is in the case of light waves. A com- 
plete and perfectly spherical wave as represented by WW in 
Fig. 336 would be focused at a spot the dimensions of which are 
of the same order of magnitude as the wave-length of the light. 

* A good discussion of this subject is to be found in Lummer's Photographic 
Optics, translated by S. P. Thompson, Macmillan & Co., 1900. See also Drude's 
Theory of Optics, pages 31-92, translated by Mann and Millikan, Longmans, 
Green & Co., 1902. A very complete discussion of this subject by Czapski, Epp- 
stein, and von Rohr is given in the second edition of Winkelmann's Handhuch der 
Physik, Vol. VI, on Optics, pages 1-470. See also Miiller-Pouillet's Lehrbuch der 
Physik, ninth edition, Volume II, Part I (Optics), pages 443-880, by Otto Lummer. 

392 




Fig. 336. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 393 



If the incoming wave-front is not a complete sphere as shown 
in Fig. 336 but a sector of a sphere as shown in Figs. 337 and 338, 
then the focal spot is greatly enlarged by diffraction. This 
effect may be roughly described as follows: Figures 337 and 338 




M 



.^^R 



.--'' 



:^ 



Fig. 337. 

represent two lenses of the same focal length which have con- 
verted a train of plane waves into the converging spherical waves 
cc cc. Let us suppose that the wave fronts cc cc are perfectly 
spherical, or in other words that the lens action is perfect in 
both figures. A very considerable portion of the wave-energy leaks 
off, as it were, from the edges cc of the waves into the surrounding 



L' 



flrffiffi 



L' 



-R 
~R 



F' 



Fig, 338. 



region RR and produces a widening of the focal spot. This is 
especially the case in Fig. 338, because there is a great deal of 
" edge " and but little '' body " to each of the waves cc in Fig, 
jj8. As a matter of fact the focal spot is much larger in Fig. 
338 than in Fig. 337.* Figure 339 is an actual photograph of a 

* The terms used in this discussion are necessarily very crude. The mathe- 



394 



THE THEORY OF LIGHT. 



focal spot such as would be obtained in Fig. 337 or 338, but 
greatly magnified. In the original photograph three or four 
rings of light are visible around the central spot instead of only- 
one. It is evident from this discussion that focal imperfections 
due to the nature of light itself are reduced to the utmost by 
making the diameter of a lens large in comparison with its focal 
length, the lens action itself being supposed to be perfect.* 





Fig. 339- 



Fig. 340. 



284. Numerical Aperture. — The free diameterf d divided by 
the focal length / of a lens is called the numerical aperture of the 
lens. See Fig. 340. Thus the object-glass of the great telescope 
of the Lick Observatory is 3 feet in diameter {= d) and its focal 
length is 50 feet ( = /) , so that its numerical aperture is 0.06. The 
best objectives for instantaneous photography have a numerical 
aperture of about 0.25, that is the free diameter is about one 
quarter of the focal length. The numerical aperture of the best 
high-power microscope objectives is 1.4 or even 1.6. 

matical theory of diffraction of this type is very compHcated. See Preston's 
Theory of Light. 

* A very interesting and extremely clear discussion of this matter is given by 
Michelson, Light Waves and Their Uses, pages 27-30, University of Chicago Press, 

1903- 

t The free diameter of a simple thin lens is the diameter of the lens. The "free 
diameter" of a compound lens is-not so easily defined. The definition of numerical 
aperture as here given is sufficient however for present purposes. For a full dis- 
cussion of aperture see Drude's Theory of Optics, translated by Mann and Millikan, 
pages 73-92; see also Dallmeyer's Telephotography, pages 91-101. A very full 
discussion of this subject is given in Winkelmann's Handhuch der Physik, Vol. VI, 
pages 211-260, 298 and 347. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 395 

Large numerical aperture is desirable for two reasons, namely, 
(a) For the reason explained in Art. 283, when one wishes to get 
the greatest possible detail in an image as in the high-power 
microscope, and (b) In order to get a bright image, whenever a 
bright image is desirable, as in instantaneous photography. 
Large numerical aperture, however, is very undesirable in case 
of a simple lens or in case of a poorly designed compound lens, 
because certain lens imperfections become very evident with 
large numerical aperture. Thus every photographer knows that 
to get sharp detail in his photograph he must reduce the free 
diameter of his lens by means of a metal diaphragm with a hole 
in it, a ''stop" as he calls it. 

The brightness of the image which a lens forms of a given distant 
object is proportional to the square of the numerical aperture of the 
lens. Consider two lenses A and B of the same focal length, 
the diameter of lens A being twice as great as the diameter 
of lens B (numerical aperture twice as great). These two 
lenses form images of the same size of a given distant object, and, 
inasmuch as the larger lens gathers four times as much light, its 
image is four times as bright. Consider two lenses A and B 
of the same diameter, lens A having a focal length twice as great 
as lens B (numerical aperture half as great) . Under these con- 
ditions the lenses gather the same amount of light, but lens A 
forms an image twice as large in diameter or four times as large 
in area as that which is formed by lens B, so that the image 
formed by lens A is one quarter as bright as the image formed 
by lens B. 

285. Field Angle. — The angle between lines drawn from the 
center* of a lens to the extreme edges of the largest distinct image 
which the lens can produce is called the field angle of the lens. 
Thus the angle F is the field angle of the lens in Fig. 341. A 
simple lens usually gives an image so badly blurred that one can 
scarcely speak of any distinct image at all, and such a lens 
becomes extremely unsatisfactory for field angles greater than a 

* This statement is made as if we were dealing with a simple lens. 



396 THE THEORY OF LIGHT. 

few degrees. The field angle of high-grade telescope and micro- 
scope objectives is ordinarily very small, seldom exceeding one or 
two degrees. Some photographic objectives, on the other hand, 

are made which give excellent 
^^. ^ definition with a field angle as 




iF 



great as 135 degrees. Such 
image ^^ ^ - 

lenses are called wide angle 



A 



lenses. 
Fig. 341. \t\s impossible to eliminate 

certain imperfections of a lens 
which is used as a wide angle lens, and therefore such lenses 
are always used with small numerical aperture in order to make 
the image reasonably sharp and distinct. 

286. Spherical aberration. — ^A plane or spherical wave is in 
general not plane or spherical after passing through a lens. This 
effect is called spherical aberration. There is, of course, an infinite 
variety of ways in which a wave front (or any surface) may be not 
spherical or plane, and spherical aberration is in fact an extremely 
complicated thing. 

A very narrow pencil of rays passing through a lens parallel 
to the axis of the lens is not subject to spherical aberration be- 
cause, in the first place, the transmitted waves are symmetrical 
about the axis (equal curvature in every direction), and, in the 
second place, the transmitted waves are so small that they cannot 
be distinguished from sectors of a spherical surface. 

A hroad beam of parallel rays parallel to the axis of a lens is 
subject to spherical aberration to a very considerable degree as 
described in the next article. 

A narrow pencil of rays which passes obliquely through a lens 
becomes a well-defined astigmatic pencil, as described in Art. 288. 

A broad oblique pencil or beam of rays is very greatly confused 
by a simple lens, as described in Art. 289. 

There are, therefore, three kinds of spherical aberration, 
namely, (a) spherical aberration of a broad pencil or beam of rays 
parallel to the axis of the lens, (6) spherical aberration of a narrow 



LENS IMPERFECTIONS AND THEIR CORRECTION. 397 

oblique pencil of rays, and (c) spherical aberration of a broad 
oblique pencil or beam of rays. The first is called axial spherical 
aberration, the second is called astigmatism, and the third is called 
oblique spherical aberration, or coma. 

287. Axial spherical aberration. — Figure 342 is a photograph 
of a cloud of white smoke as illuminated by a beam of parallel 




Fig. 342. 

rays after passing through the lens as indicated by the arrows 
in the figure. The outline of the lens can be seen faintly at the 
left in the photograph which is entirely untouched except for 
the arrows and the lettering. The light which passes through 
the middle part of the lens is focused at F, the light which passes 
through the edge of the lens is focused at F\ and the light 



t 


\^ 




axis of ^\^ 




iem ^^ 


L 


^Iz 



?<'\""' 


■ 




•W^/ 






^:^-- 


-^•^ 





Fig. 343. 



398 



THE THEORY OF LIGHT. 



which passes through the intermediate parts or zones of the lens 
is focused along the bright line between F and F' . To fully 
understand the details of Fig. 342 let us consider the sketches 
which are shown in Figs. 343 and 344. Figure 343 shows the 




Fig. 344- 

rays v^^hich pass through the zone ZZ of the lens (the zone which 
lies between the dotted circles in Fig. 343) ; and Fig. 344 shows 
a side view of a narrow pencil of rays PCD. This pencil becomes 
an astigmatic pencil after passing through the lens; one focal Hne 
of the astigmatic pencil is at C (perpendicular to the plane of 
the paper) and the other focal line of the pencil is the short 
portion DD of the axis of the lens. The curved line in Fig. 344 
(which, of course, represents a surface) is a caustic and it is 
visible in Fig. 342 as the sharp boundary of the illuminated 
cloud of smoke. 

288. Astigmatism. — A narrow pencil of parallel rays, or indeed, 
any narrow homocentric pencil of rays, becomes an astigmatic 
pencil when it passes obliquely through a simple lens. Thus, 
Fig. 345 shows the focal lines C and DD of the astigmatic 
pencil of rays which is produced by the passage of the narrow 
pencil rr obliquely through the lens, as shown. The focal point 
of the lens is at 7^. This conversion of a homocentric pencil of 
rays into an astigmatic pencil by oblique passage through a lens 
is called astigmatism, and a compound lens which is free or 
approximately free from astigmatism is called an anastigmatic, 
or simply a stigmatic lens. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 399 

A very simple demonstration of the astigmatism of a simple 
lens may be made by looking obliquely through a magnifying 
glass at the cross-rulings on a sheet of cross-section paper. 




Fig. 345. 

Under these conditions one or the other set of cross-rulings may 
be sharply focused according to the distance of the lens from the 
paper. 

A more striking demonstration of astigmatism may be made 
by projecting a cross-ruled lantern slide upon the screen with 
the lantern objective turned so that the light passes through it 
obliquely. For this experiment use a simple lens as the lantern 
objective and cover all but the central portion of the lens with 
opaque paper. 

A ^ 




^. 



^/^ 



% 



^JfF 



\ 



Fig. 346. 



400 



THE THEORY OF LIGHT 



Astigmatism of the eye is due to unequal curvatures of the 
refracting surfaces of the eye in different directions so that a 
homocentric pencil of rays which enters the eye axially becomes 
an astigmatic pencil. This kind of astigmatism must not be con- 
fused with the astigmatism of a lens which is the conversion of 
an oblique homocentric pencil into an astigmatic pencil. 

289. Oblique spherical aberration of a broad beam or pencil. 
Coma. — Figure 346 is a sketch, exactly one quarter size, showing 
a beam of parallel rays passing obliquely through a simple con- 




Fig. 3470. 



Fig. 347&. 



verging lens LL. Figure 347^^ is a photograph full size of the 
spot of light on the plate AB oi Fig. 346, and Fig. 347& Is a 
photograph of the spot of light on the plate A'B' of Fig. 346, the 
full opening of the lens LL being used. 

When the central zone only of the lens LL Is used, the astig- 
matic pencil is focused along a horizontal line at C (perpen- 
dicular to the plane of the paper) and along the vertical line DD 
In Fig. 346, as shown by the two sketches In Fig. 348. In these 
sketches the outline of the coma is also shown for the sake of 
comparison. 





Fig. 348a. 



Fig. 3486. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 40I 

290. Compensation for spherical aberration. Aplanatism. — 

Lens imperfections are always eliminated (approximately) by 
balancing, as it were, the opposite imperfections of two or more 
simple lenses which are used together as a compound lens.* 
The possibility of eliminating axial spherical aberration in this 
way was discovered about 1760, and the first step towards the 
correcting of a compound lens for oblique spherical aberration 
w^as made by Frauenhofer about 18 10. A more complete solu- 
tion of the problem of oblique spherical aberration was made by 
Abbe in 1873 when he discovered what is known as Abbe's sine 
condition which refers to slightly oblique rays passing through a 
very wide aperture lens. The possibility of correcting a com- 
pound lens for astigmatism, which refers to very oblique rays 
through a lens of comparatively narrow aperture, was discovered 
by Rudolph about 1890.! 

Many of the earlier lens manufacturers had learned by trial to 
satisfy Abbe's sine condition, but since the diwscoveries of Abbe 
and Rudolph, the lens designer recognizes more distinctly the 
necessity and understands more clearly the possibility of elimi- 
nating astigmatism and coma, but the lens designer must still 
work very largely by trial because of the infinite number of pos- 
sible combinations of differently shaped lenses of different kinds 
of glass at different distances apart. 

* This is true except in the case of very large telescope objectives where it is 
only partially true. Some of the imperfections of a large telescope objective are 
eliminated by bringing one or more of the lens surfaces to a desired shape (other 
than spherical) by very tedious local polishing. 

t An example of the complete calculation of a telescope objective, according to 
Gauss, is given by Otto Lummer in Miiller-Pouillet's Lehrbuch der Physik, Vol. II, 
part I, pages 573-579. The method for eliminating spherical aberration is brought 
out in this example. 

The theory of the aberrations of a lens was worked out very completely by von 
Seidel in 1855. A good discussion of von Seidel's theory may be found in Lummer's 
Photographic Optics, translated by S. P. Thompson, pages 6-13, and pages 103-115; 
and a good discussion of Abbe's sine condition may be found on pages 1 16-12 1. See 
also Drude's Theory of Optics translated by Mann and Millikan, pages 58-63. An 
extremely simple discussion of the five aberrations of von Seidel on the basis of 
Hamilton's Principle is given by Lord Rayleigh in the Philosophical Magazine for 
June, 1908, pages 677-687. 
27 



402 



THE THEORY OF LIGHT. 



A lens which is free from spherical aberration is said to be 
aplanatic. This term was originally applied to a lens which 
had been corrected only for axial spherical aberration, but now- 
adays we must distinguish two cases as follows: 

(a) A very wide aperture lens may be corrected for axial 
spherical aberration and for the aberration of a very slightly 
oblique broad beam. Such a lens is said to be aplanatic; but it 
is aplanatic only for one particular pair of conjugate points 
which are called the aplanatic points of the lens. 

ih) A narrow aperture lens may be corrected for the spherical 
aberration of a very oblique narrow pencil (astigmatism). Such 
a lens is said to be anastigmatic. 

291. Image distortion. — ^The image of an object formed by a 
lens is in general distorted. Thus, Fig. 349 represents a square 
network of lines, and Figs. 350 and 351 show two distorted 
images of this network. In Fig. 350 the magnification of the 






Fig. 349. 



Fig. 350. 



Fig. 351. 



image is less near the edges of the field of view than it is near the 
center, and in Fig. 351 the magnificadon of the image is greater 
near the edges of the field of view than it is near the center. 

The distortion of an image by a lens may be shovvn in a very 
striking wa^^ as follows: A magic lantern is provided with a simple 
objective lens LL, Figs. 352 and 353, and the image of a cross- 
ruled lantern slide SS is projected upon the screen. 

{a) The source of light in the lantern is adjusted so that the 
light, after passing through SS, is concentrated at B, Fig. 352, 
as if the point B were a small hole in a diaphragm in front of the 



LENS IMPERFECTIONS AND THEIR CORRECTION. 403 



object lens LL. In this case the cross-rulings, as projected on 
the screen, appear Hke Fig. 350. 





Fig. 352. 

(6) The source of Hght in the lantern is then adjusted so that 
the light, after passing through ^S is concentrated at the point 
B, Fig. 353, as if the point B were a small hole in a diaphragm 
behind the object lens LL. In this case the cross-rulings, as 
projected on the screen, appear like Fig. 351. 





,..- to screen 

B ^' 



Fig. 353. 

These experiments show that the distortion of an image is of 
one kind or the other (like Fig. 350 or Fig. 351) according as the 
diaphragm, or stop, B is in front of or behind the lens, and one is 
able to see in a general way that one kind of distortion may be 
compensated by the other kind by placing the diaphragm between 
two lenses of a system, as shown in Fig. 354, in which the dia- 
phragm, or stop, DD is behind one lens LL, and in front of 
the other lens L'L\ 

The action of the symmetrical double-lens of Fig. 354 may be 



404 



THE THEORY OF LIGHT. 



understood from the following considerations. The hole H 
in the diaphragm is assumed to be small so that all incident 
rays like R which come from points in the object plane intersect 
at the point x which is conjugate to the point H with respect 
to the front lens LL, and all emergent rays like R' intersect 
at the point y which is conjugate to the point H with respect 
to the back lens L'U . Furthermore every incident ray R is 



__oxisof 
X system 




image plam 



parallel to the corresponding emergent ray R\ Therefore the 
points like b in the image plane as seen from the point y are 
distributed in exactly the same way as the corresponding points 
like a in the object plane as seen from the point x, or, in other 
words, the image is exactly similar to the object.* 

A lens system which gives an undistorted image of an object 
is called a rectilinear or orthoscopic system. Such lenses are 
always used for photo-engraving, where it is desired to produce 
an accurate copy of a drawing, and for photographing buildings. 

292. Curvature of field. — In order to project upon a screen the 
most distinct image of an extended object which it is possible to 
form by means of a simple lens, the screen must be curved as 
shown by 55 in Fig. 355. It is instructive to note the relation 
between Figs. 345 and 355. There is, between C and DD in 
Fig. 345, a place where the astigmatic pencil would produce the 

* This discussion ignores the spherical aberration of the lenses LL and L'L' 
in Fig. 354. The theory of the orthoscopic lens is quite fully discussed on pages 
29-39 oi Lummer's Photographic Optics, translated by S. P. Thompson. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 405 

smallest possible luminous spot, the circle of least confusion, so- 
called, and the screen 55 in Fig. 355 must pass between C 
and DD of Fig. 345. 

The imperfection of a lens which is here described is called 




Fig. 355. 

curvature of field and a lens system in which this error is cor- 
rected is said to have o. flat field* 

293. Chromatic aberration. — ^The five lens errors, axial spher- 
ical aberration, astigmatism, coma, image distortion and curva- 
ture of field, refer to the action of a lens when light of one wave- 
length (one color) is used. The use of white light introduces 
another complicated imperfection which is called chromatic aberra- 
tion and which is due to the fact that a given sample of glass has 
different refractive indices for different wave-lengths (colors) of 
light. Thus, Fig. 356 shows the focal points v and r of a lens 
LL for violet light and for red light respectively; the focal points 
for the other colors lie between v and r. The smallest diameter 
of the focal spot ah, Fig. 356, is about one thirty-third of the 
diameter of the lens. 

The distance vr, Fig. 356, for a lens of given focal length 
varies greatly with different kinds of glass, and it is therefore pos- 
sible to construct a converging lens of one kind of glass and a 
diverging lens of another kind of glass so that when the two 

* Curvature of field is intimately connected with astigmatism although von 
Seidel's theory gives two distinct conditions, one for the elimination of astigmatism 
and another for the elimination of curvature of field. . See Winkelmann's Handbuch 
der Physik, Vol. VI, pages 139-143. Curvature of field is disscused on pages 10, 23, 
57 and 61-67 of Lummer's Photographic Optics. 



406 THE THEORY OF LIGHT. 

lenses are used together as a compound lens system the chromatic 
aberration of one of the lenses is annulled (nearly) by the oppo- 
site chromatic aberration of the other, while the converging action 
of the one is not annulled by the diverging action of the other. 
A compound lens which is compensated for chromatic aberration 



Fig. 356. 

in this way is said to be achromatic. The achromatic doublet 
was devised in 1758 by Dolland, who used such doublets for 
the object-glasses of his telescopes. A sketch of the theory of 
the achromatic doublet is given in the chapter on dispersion. 

Some idea of the complexity* of chromatic aberration may be obtained by the 
following discussion of Figs. 357, 358 and 359. A lens system may be "achro- 
matized" (a) so as to form images of all colors'\ in one plane, or (&) so as to provide 
for equal sized images of all colors.'^ In the first case the different colored images 
will be of different sizes, and in the second case the different colored images will 
be in different planes. The first is called the achromatization of the focal planet 
and the second is called the achromatization of magnification.^ 

Consider a converging lens CC and a diverging lens DD arranged as shown 
in Fig. 357. Let us for a moment ignore chromatic aberration and consider that 
the two lenses CC and DD focus a beam of parallel rays at the point F. Extend 
the two rays Fx backwards until they intersect the incident rays RR at LL. 
The combined action of the two lenses CC and DD is equivalent to a single lens at 
LL of which the focal length is equal to f as shown. 

Let CC, Fig. 358, represent a converging lens of crown glass and DD a di- 
verging lens of flint glass, the two lenses being designed to bring both red light r 
and violet light z' to a focus at the point F as shown. The two lenses CC and 
DD, in so far as their action on red light is concerned, are together equivalent to 
a single lens at A of which the focal length is fr as shown; and the two lenses 

* This statement does not refer to the impossibility of compensating chromatic 
aberration for all colors. 

t Strictly, two colors only.' 

t Sometimes called achromatization of the focal point. 

§ Sometimes called achromatization of the focal length. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 407 

are equivalent to a single lens at B (focal length fv) in so far as their action on 
violet light is concerned. The lens system CCDD of Fig. 358 produces red and 




Fig. 357- 

violet images of a distant object in the same plane, namely, the plane containing 
the point F, but the violet image is larger than the red image in the ratio of fv 
to fr- The lens system CCDD, Fig. 358, is achromatized for focal point but not 
achromatized for focal length. 

Figure 359 represents a combination of a converging crown glass lens and a 
diverging flint glass lens which is achromatized for focal length {fr = fv)- Such 
a system would give the same sized violet and red images of a distant object, but 
the images would not be in the same plane. In fact the red image would be in 
the plane containing Ft and the violet image would be in the plane containing F^. 

When the two lenses CC and DD are very near together (near, that is, in 
comparison with their focal lengths), then achromatization of the focal point carries 



If. 

i 






K— - 






A ~~'-~^^ 



axis of, 




converging lenaV. 
"^" of crown glassy. 



Fig. 358. 



4o8 



THE THEORY OF LIGHT. 



with it the achromatization of focal length, as is evident from a careful study of 
Figs- 358 and 359. Complete* achromatization is therefore much more easily 




Fig. 359. 

accomplished when the individual lenses of a system are near together than when 
they are far apart. In photographic objectives it is usually desirable to separate 
the individual lenses, and in such cases achromatic doublets are used instead of 
simple individual lenses. 

A lens system which consists of two lenses of similar glass at a distance apart 
equal to half the sum of their individual focal lengths has the same focal length 
for all wave-lengths (all colors). In this case, however, the lenses are at a great 
distance apart as compared with their focal lengths, and, although such combina- 
tions have the same focal length for all wave-lengths, they do not have the same 
focal point for all wave-lengths. Such a combination is therefore achromatized 
for focal length but not achromatized for the focal point. The doublets of Huygens 
and Ramsden which are so much used for eye-pieces for telescopes and microscopes 
are examples of partially achromatized systems of this type. 

Chromatic dififerences of spherical aberration. — A lens system may be accurately 
aplanatic (with respect to its aplanatic points, of course) for a given wave-length 
(color) of light and non-aplanatic for other wave-lengths (colors). This error of a 
lens is called the chromatic difference of spherical aberration. 

294. The wide-angle lens. — A lens in the form of a complete 
sphere having a diaphra.gm DD with a small hole at its center, 
as shown in Fig. 360, has a field angle of nearly 180°, but the 
field is very strongly carved, as shown. The wide-angle photo- 
graphic objective involves the principle of the spherical lens, 

* Complete in the sense of including achromatization of focal point and achroma- 
tization of focal length, not in the sense of being achromatized for all colors of the 
spectrum. 




LENS IMPERFECTIONS AND THEIR CORRECTION. 409 

modified more or less to give flatness of field. This is exemplified 
by Fig. 369 which is a moderately wide-angle photographic 
objective and by Fig. 372 which is an extremely wide-angle 
photographic objective. 

295. Wide-angle lenses versus wide-aperture lenses. — A lens 
cannot be made to give a wide field-angle and to have at the same 
time a large numerical aperture. 
The combination of these two 
things is impracticable. When a 
very wide field-angle is desired one 
must he content with small aper- 
ture, and when a very wide aper- 
ture is desired one must he content 
with small field-angle. Thus, a 
high-grade microscope objective 
with a numerical aperture of about Fig. 360. 

1.4 has a field-angle of not more 

than half a degree, and a wide-angle photographic lens having 
a field-angle of no degrees has a numerical aperture of about 

1/36. 

There is a demand in photography, however, for lenses having 
a moderately wide aperture and giving a moderately wide field, 
and many photographic lenses are now available which give 
fairly good definition over a field from 30 to 60 degrees wide with 
numerical aperture ranging from 1/4 to 1/8. See Figs. 370 and 

371- 

A wide-aperture lens must he, ahove all things, compensated for 
spherical aher ration and for chromatic aherration. — The effect of 
spherical aberration In producing very great blurring at the focus 
of a large-aperture lens as compared with the blurring at the focus 
of a small-aperture lens Is shown In Fig. 361. The same effect 
for chromatic aberration Is shown In Fig. 362. In these figures 
SS represents the position of the screen for which the focal spot 
is the smallest possible. The distance ah In Fig. 361 Is roughly 
proportional to the square of the aperture of the lens so that a'h' 



410 



THE THEORY OF LIGHT. 



in Fig. 361 is very much less than ah, and the size of the focal 
vSpot is extremely small for lens B as compared with its size for 
lens A. The distance rv in Fig. 362 is independent of the 





Fig. 361. 

aperture of the lens, but the figure shows nevertheless that a 
beam of light is focused in a much smaller spot by lens B' than 
by lens A' . 

A wide-angle lens must he, ahove all things, compensated for 
astigmatism, distortion, and curvature of field. Most wide-angle 
lenses depend, however, upon narrowness of aperture for keeping 
these imperfections within practicable limits. This is exemplified 
by Figs. 369 and 372. 

296. Examples of compensated lens systems. — (a) Telescope 
objectives. — The telescope objective is generally used where a 
small field- angle is desired, and the ordinary telescope objective 
is compensated only for spherical aberration and for chromatic 



LENS IMPERFECTIONS AND THEIR CORRECTION. 411 



aberration. Figure 363 shows a sectional view, actual size, of an 
aplanatic achromatic telescope object-lens of 18 inches focal 




C P 



Fig. 362. 

length as designed by Fraunhofer. It consists of a double convex 
crown glass lens CC and a concavo-convex flint glass lens FF. 
Figure 364 is a sectional view, actual 
size, of a modern high-grade opera-glass 
of which the object-lens and the 
eye-lens E are triplets. The object- 
lens of an opera-glass is much larger in 
numerical aperture (larger in diameter 
for a given focal length) than the object- 
lens of the astronomical telescope as 
usually constructed. It is for this 
reason desirable to use three lenses for 
the object-lens of an opera-glass, as shown in Fig. 364, in 
order that the various imperfections may be more completely 




412 



THE THEORY OF LIGHT. 



compensated than would be possible by the use of two lenses, as 
shown in Fig. 363. 



Scr^*^ 



crown 




crown 



oxu 



crowi 




crown 



ttmt 



Fig. 364. 

{b) Eye- pieces * — Figure 365 shows an eye-piece doublet which 
was designed by Huygens about 1680. It is partially achromatic 
(achromatized for focal length but not for focal points) and it 



from ^ 

object 
glass 




Fig. 365. 
Huygens eyepiece. 

has a wide flat field. It is extensively used at the present day 
with compound microscopes and telescopes where it is not desired 
to use cross- wires in the image plane of the object-glass. 

Figure 366 shows an eye-piece doublet which was designed by 
Ramsden in 1783. It is partially achromatic (achromatized for 
focal length but not for focal points) and It has a wide flat field. 
It Is extensively used at the present day with compound micro- 
scopes and telescopes where it is desired to use cross-wires in the 
image plane of the object-glass as shown in Fig. 335. 

* A simple discussion of eyepieces is given in Edser's Light for Advanced Students, 
pages 204-212. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 413 

Figure 367 shows a highly perfected achromatic and ortho- 
scopic eye-piece designed by Abbie. 




from 
object 
glass 



Fig. 366. 
Ramsden eye piece. 



(c) Photographic lenses. "^ — The first step in the development 
of the modern high-grade photographic lens was taken by Wol- 
laston in 18 12 who substituted the meniscus lens (see Fig. 301) 
for the simple bi-convex lens which was used in the camera 



>£ttnt 




from thject glas» 




cromt 



CTQUm 



Fig. 367- 
Abbe eyepiece. 

obscuraf before his time, and he specified a particular position 
for the diaphragm or stop. The simple meniscus lens is exten- 
sively used at the present day in cheap forms of photographic 
cameras. 

The achromatic doublet came into use for the camera obscura 

* A good outline of photographic optics is given by Lummer in the Zeitschrift fiir 
Instrumentenkunde, 1897, pages 208, 225 and 264. These articles have been trans- 
lated into English by S. P. Thompson and published as Photographic Optics by Mac- 
millan & Co., 1900. A very complete discussion of the theory and history of the 
photographic objective is von Rohr's Theorie und Geschichte des Photographischen 
Objectivs, Berlin, 1899. See also Photography for Students by Louis Derr, The 
Macmillan Company, 1906. 

t The camera obscura is a dark chamber, or box, with a lens at one side for pro- 
jecting an image of an external object or landscape. 



414 



THE THEORY OF LIGHT. 



about 1835. This lens has been and is still extensively used for 
photographic purposes. 

The demand for a quick-acting lens (wide numerical aperture) 
which came with the invention of photography was met by the 




I 



D 



axis 



-dint 




crown 



eroumi 



flint 

I" 

Fig. 368. 
Petzval's portrait lens. 

remarkable portrait lens of J. Petzval which was designed in 1840 
and manufactured by Voigtlander. Figure 368 shows a sectional 

view, full size, of a Petzval por- 
trait objective having a focal 
length of 10 centimeters. This 
lens has a numerical aperture of 
1/3.5 ^iid it is accurately aplan- 
atic. It gives extremely good 
definition in the center of its field 
and fairly good definition over 
a field of about 25 or 30 degrees. 
This objective and the portrait 
objective of J. H. Dallmeyer, which was brought out in 1866, 
are still extensively used. 

After the remarkable achievement of Petzval, the next notable 
improvement in the photographic lens was made in England, 
and the first symmetrical orthoscopic lens was the "Globe lens" 
of Harrison and Schnitzer, which was brought out about i860. 
Figure 369 is a sectional view, full size, of a "Globe lens" of 10 
centimeters focal length. 




crown 



Fig. 369. 
The "Globe lens. 



i^UM 



LENS IMPERFECTIONS AND THEIR CORRECTION. 415 



In 1866 the symmetrical aplanatic orthoscopic lens of Steinheil 
was produced. A sectional view, full size, of one of these lenses 
of 20 centimeters focal length, is shown in Fig. 370. 



Hint 




I 



crown 



n 



axis 



crown 



V 




flint 



Fig. 370. 
Steinheil' s lens. 

All converging lenses in Figs. 368, 369 and 370 are of crown 
glass and all diverging lenses are of flint glass. The compensa- 
tion of lens errors depends upon the combination of lenses made 
of different kinds of glass, and the de- 
velopment of photographic lenses (and 
also of microscope objectives) was greatly 
stimulated after the establishment of the 
celebrated Jena Glass Works in 1885 for 
the manufacture of a variety of new 
optical glasses under the direction of 
Abbe. 

Figure 371 shows one of the best of 
present day photographic lenses and 
with an aperture of 1/4.5. It is very 
completely corrected for axial and 
obHque spherical aberration, it is very 
nearly orthoscopic and it gives a flat 
field. 

The symmetrical achromatic doublet which is shown in Fig. 
369 gives good definition over a field of 90° or more, but Fig. 372 




Fig. 371. 
" Tessar Lens. " Carl 
Zeiss. Patented 1902 by P. 
Rudolph. 



4i6 



THE THEORY OF LIGHT. 



shows an extremely wide-angle lens which gives good definition 
(with a small stop) over a field of 135°. This lens is interesting 
in that it illustrates the fact that a very nar- 
row aperture lens does not need especially to 
be corrected for spherical and chromatic aber- 
ration as explained in connection with Figs. 
361 and 362. 

For making photographs of a distant small 
object, a very long focal length lens is neces- 
sary if a fair-sized photograph is to be pro- 
duced. In order to avoid the great incon- 
venience which would be involved in the use 
of a very long focal length lens of the ordi- 
nary type, a combination called the telephoto- 
graphic lens is used. Figure 373 shows a tele- 
photographic lens. The equivalent focal 
length of this lens may be varied by changing 
the distance between the front and back com- 
binations, and one may easily obtain an 
equivalent focal length of 8 or 10 feet with 
a camera length of 2 or 3 feet. The action of 
the telephotographic combination may be un- 
derstood with the help of Fig. 357. 

(d) Microscope objectives. — Low-power mi- 
croscope objectives usually consist of one 
achromatic doublet. Figure 374 shows a low- 
power microscope objective consisting of two 
achromatic doublets combined to form one system. 

In high-power microscope objectives, however, the front lens of the 
object-glass (the lens nearest to the object) is always made in the 
form of a hemisphere, and in the most powerful microscope objec- 
tives the space between the front lens and the object ivhich is being 
examined is filled with oil {the oil having the same index of refraction 
as the front lens of the object-glass) as shown in Fig. J75. A micro- 
scope objective when so used is called an oil immersion objective. 




Fig. 372. 
"Hypergon Lens.* 
Patented 1900 by C 
P. Goerz. 



LENS IMPERFECTIONS AND THEIR CORRECTION. 417 

Figure 375 shows (three times actual size) a highly perfected 
microscope objective of 2 millimeters (one twelfth inch) focal 




Fig. 373- 
"Telephoto Lens." Patented about 1891 by T. R. Dallmeyer. 

length. This objective was designed about 1886 by Abbe and 
it is known as the apochromutic objective. It has the largest 
possible numerical aperture, and, although it is corrected only 
for spherical aberration and for chromatic aberration, it involves 
eleven separate compensating effects. It is achromatized for 
focal point for three colors, and its spherical aberration is elimi- 
nated for each of two colors for slightly oblique rays as well as 




WiJi// 





Fig. 375- 

for rays parallel to the axis. The lack of achromatization of focal 
length causes the objective to form colored Images of different 
sizes, and this imperfection is compensated by using a specially 

28 



41 8 THE THEORY OF LIGHT. 

designed non-achromatic eye-piece which magnifies the different 
colored images differently thus giving a single colorless resultant 
image. The excellence of this objective is such that the absolute 
limit of effective magnifying power as fixed by the nature of light 
(as explained briefly in Art. 283) may be considered as actually 
attained by it. 



CHAPTER XXIII. 

DISPERSION* SPECTRUM ANALYSIS.f 

297. Newton's experiment. { Homogeneous light. Non- 
homogeneous light. — A beam of parallel rays of white light, 
such as sun light or lamp light, is changed into a fan-like beam by 
passage through a prism. Thus, the beam of parallel rays B, 
Fig. 376, is changed into the fan-like beam B' by passage 
through the prism P. This fan-like beam in falling upon a 
screen 55 produces an illuminated band R V called a spectrum 
which is red at the end R and passes by insensible gradations 

* A good discussion of dispersion is given in Edser's Light for Students, pages 
375-387, See also Wood's Physical Optics, pages 85-99 and 308-348. The 
modern theory of dispersion is given in Drude's Theory of Optics (translated by 
Mann and Millikan), pages 382-399. 

The rainbow is produced by the dispersion of sun light in drops of rain. Other 
interesting optical phenomena of the atmosphere are mirage, coronas and halos, 
scintillation and the color of the sky. These various matters are discussed in 
Hastings' Light, pages 111-153 (Scribner's, 1901); Preston's Theory of Light, 
pages 529-541 (Macmillan Sz: Company, 1901); Edser's Light for Students, pages 
101-107 (Macmillan & Company, 1902); and Wood's Physical Optics (The Mac- 
millan Co., 1905), pages 69-78. Wood's discussion of mirage is especially interest- 
ing. The sharp-edged appearance of the sun is a mirage effect. See A. Schmidt, 
Physikalische Zeitschrift, Vol. IV, pages 282-285, February, 1903. 

t A very good simple discussion of spectrum analysis and its teachings is to be 
found on pages 330-360 of Edser's Light for Students, Macmillan, 1902. One of 
the best small works in English on spectrum analysis is Spectroscopy by E, C. C. 
Baly, Longmans, Green & Co. See also Landauer's Spectrum Analysis translated 
by J. Bishop Tingle, John Wiley & Sons. A very interesting discussion of the appli- 
cation of the interferometer to spectroscopy is given on pages 60-83 of Michelson's 
Light Waves and their Uses, University of Chicago Press, 1903. The interesting 
effect of magnetic field upon the spectrum of a hot gas (the Zeeman effect) is dis- 
cussed by Michelson on pages 107-126. 

The most complete work is Kayser's Spectralanalyse, and perhaps the best resume 
is Kayser's chapter (pages 654-784 of volume VI) in Winkelmann's Handbuch der 
Physik. 

% An interesting account of Newton's original experiments with the prism may be 
found in Preston's Theory of Light, pages 1 17-122. 

419 



I 



420 



THE THEORY OF LIGHT. 




through orange, yellow, green, and blue to violet at the end V. 
The beam B of white light is said to be dispersed by the prism. 
A photographic plate reveals the existence of invisible rays 
beyond V. These rays are called the ultra-violet rays. A heat 
detector shows the existence of rays below R. These rays are 
called infra-red rays. The portion of the spectrum between R 
and V is called the visible spectrum. 

The beam of white light B, Fig. 376, is deflected by the prism 
and also spread out or dispersed. On the other hand, the beam 

of light B" , which passes through 
a small hole in the screen, can 
be deflected by a prism but it can- 
not be spread out or dispersed. 
The beam of white light is evi- 
dently made upof dissimilar parts 
because these parts are unequally 
deflected by a prism. Therefore, 
white light is called non-homo- 
geneous light. The beam B'\ 
on the other hand, consists of but one kind of light because it is 
deflected by the prism without being dispersed. Therefore, the 
beam B'^ is called homogeneous"^ light. Since the prism P 
deflects the different parts of the non-homogeneous beam B 
differently, it is obvious that the glass of which the prism is 
made has a different index of refraction for each of the homo- 
geneous parts or components of white light. 

The phenomena of interference which are described in the next 
chapter show that a beam of homogeneous or monochromatic light 
is a simple wave-train of definite wave-length or, more strictly 
speaking, a simple wave-train of definite frequency, inasmuch as 
the wave-length is halved, for example, when a given wave-train 
passes into a medium in which the velocity is halved. A beam 
of white light, on the other hand, is an utterly irregular succession 
of wave-pulses and short sections of wave-trains of every variety of 

* Homogeneous light is sometimes called monochromatic light, or light of one 
color. 



Fig. 376. 



t==s= 



oar 




water 



DISPERSION. SPECTRUM ANALYSIS. 42 1 

wave-length. In a vacuum, all these different waves travel at the 
same velocity, and this is approximately true in air also. In sub- 
stances like glass, wave-trains of different wave-lengths have dis- 
tinctly different velocities and therefore the different wave- trains 
which make up a beam of non-homogeneous light are differently 
refracted by a glass prism. f 

The velocity of ordinary water waves, especially of the variety 
known as ripples, varies greatly with the wave-length, and a 
phenomenon which is closely analogous to the dispersion of light 
by a prism (due to the different velocities of the wave-trains of 
lightof differentwave-lengths) 
is the following: An oar is 
dipped gently into the smooth 
surface of a pond and the ir- 
regular wave which is pro- 
duced by the oar is quickly 
resolved into a series of fine Pig 3^^, 

ripples, as shown at A, Fig. 

377. The fine ripples of short wave-length travel faster than the 
coarse ripples of long w^ave-length and are thus separated from 
them, as shown. 

298. The spectroscope. — In the spectrum as obtained by 
Newton (see Fig. 376) , the beam of white light has some breadth 
and the various beams of homogeneous light into which the white 
light is resolved by the prism are each as wide as the original 
beam of white light. Therefore the various beams of homo- 
geneous light overlap each other greatly, that is to say, each point 
on the screen in Fig. 376 is illuminated by several overlapping 
beams of homogeneous light. The spectroscope is an instrument 
for separating as completely as may be the homogeneous com- 
ponents of a beam of non-homogeneous light. 

The spectroscope is exemplified in its simplest form by placing 
a large prism P in front of the object-glass of a telescope 

t The action of a prism in resolving a beam of white light into a series of simple 
wave-trains is a phenomenon of resonance. See Drude's Theory of Light (translated 
by Mann & Millikan), pages 382-399. 



422 



THE THEORY OF LIGHT. 



OE, as shown in Fig. 378; the light from a star is then deflected 
by the prism and appears to have come from X. In fact, the 
different wave-lengths of light from the star are differently 
deflected and the light appears to have come from a number of 
stars near X, one star for each wave-length of light. The result 
is that the object-glass O of the telescope forms a row of images 
of the star at RVj one image for each wave-length. This row 
of images constitutes the spectrum of the star and it may be 
examined by the magnifying glass (eye-piece) E. 

If the attempt is made to use the arrangement shown in Fig. 
378 for looking at a point-source of light near at hand instead of 



^ 

^ 







f^* •••••••.• , jj 





looking at a star, then the images at VR will not be sharply 
defined because the light from the near source S will enter the 
prism P as a series of spherical waves, as shown in Fig. 379; 



DISPERSION. SPECTRUM ANALYSIS. 



423 



these spherical waves are refracted at the plane surface of the 
prism subject to spherical aberration as explained in Art. 264 and 
as represented in Fig. 298; and the result is that the waves 
which emerge from the prism P are non-spherical and they 
cannot be sharply focused by the lens 0. In order to overcome 
this difficulty a lens is placed between the point-source 5 and 
the prism P in Fig. 379, the point-source 6* being at the focal 
point of the lens. With this arrangement, which is shown in 
Fig. 380, the beam of divergent rays from S is converted into a 
beam of parallel rays by the lens (that is, the spherical waves 
from 5 are converted into plane waves by the lens) and this 
beam of parallel rays (plane waves) is refracted by the prism 
without spherical aberration. 

The light which is to be analyzed passes through a very nar- 
row slit 5, Fig. 380, between two straight metal edges. This 
slit is then, in effect, the source of the light, and it is at the 



\ 




\ 




Fig. 380. 



focal point of an achromatic lens C, which is called the collimating 
lens of the spectroscope. After passing through the collimating 
lens, the light passes through the prism P and is then focused 
by the lens at a series of points (images of the slit) at RV. 
These images of the slit are examined by the magnifying glass 
(eye-piece) E. The band of images at RV is csXX^d tho^ spectrum 
and the individual images of the slit are called the lines of the 
spectrum. The slit 6" and the lens C are mounted at the ends 
of a short tube which is called the collimator, and the lens 



424 



THE THEORY OF LIGHT. 



and the eye-piece E are mounted at the ends of a tube which is 
csW^dthe telescope. The eye lens E is always a compound lens; 
usually like Fig. 366. 






n 



Fig. 381. 

The usual arrangement of the spectroscope is shown in Fig. 
381 in which two independent sources Si and ^2 are arranged 
to send light through the slit by covering one end of the slit 
with a total reflecting prism so that light from ^i passes through 
the upper end of the slit and light from ^2 passes through the 
lower end of the slit. The two spectra (the spectrum of Si and 
the spectrum of ^2) are then seen one above the other in the 
spectroscope and they may be compared with great ease. The 
total reflecting prism is often called the comparison prism. 

In order to be able to read off the positions of the images of 
the slit at RV, a lamp S^ is arranged to send light through a 
transparent scale AB which is at the focal point of the lens L. 
The light which passes through the scale AB is reflected from 



DISPERSION. SPECTRUM ANALYSIS. 



425 



the face of the prism into the telescope lens 0, and an image of 
the scale AB is formed in the plane i^F so that the positions 
of the images of the slit may be read off. A general view of a 
spectroscope is shown in Fig. 382. 




Fig. 382. 

299. Continuous spectra.— When light from a hot solid or 
liquid is analyzed by a spectroscope, a continuous band of images 
of the slit is produced at RV, Fig. 380. Such a spectrum is 
called a continuous spectrum. Light which gives a continuous 
spectrum contains wave- trains of every gradation of wave-length. 
The candle flame, the petroleum flame, and the gas flame give 
continuous spectra. The light from such flames is given off by 
hot particles of solid carbon. 

300. Bright-line spectra. — When the light which Is emitted by 
a hot vapor or gas is analyzed by a spectroscope, a group of dis- 
tinctly separate images of the slit is produced at RV, Fig. 380. 
Such a spectrum is called a bright-line spectrum inasmuch as the 
separate images of the slit appear as bright lines. Light which 
gives a bright-line spectrum contains wave-trains of certain defi- 
nite wave-lengths only. 



426 THE THEORY OF LIGHT. 

Every gas or vapor has a characteristic spectrum, that is, a 
characteristic grouping of images of the sUt in a spectroscope. 

301. Dark-line spectra. — When an intense beam of white Hght, 
containing all wave-lengths, is passed through a relatively cool 
vapor or gas and then analyzed by the spectroscope, dark lines 
(missing images of the slit) are seen where bright lines would be 
located in the direct spectrum of the vapor or gas. That is to 
say, a relatively cool vapor absorbs those particular wave-trains 
which it would itself give off if hot. This relation between the 
bright-line spectrum of a vapor or gas and the dark-line spec- 
trum of the vapor or gas was discovered by Bunsen and Kirchofif 
in 1865. Using the flame of a Bunsen burner and charging it 
with sodium vapor by the vaporization of common salt, they ob- 
tained the ordinary bright-line spectrum of sodium. Then pass- 
ing an intense beam of white light from a lime-light through the 
Bunsen flame into the slit, it was found that the absorption of the 
sodium vapor was such as to leave relatively dark lines in place 
of the bright lines given by the flame alone. 

The most striking dark-line spectrum is the solar spectrum, 
which shows a great number of dark lines. The dark lines in 
the solar spectrum were first studied by Fraunhofer in 1819 and 
they are called Fraunhofer s lines. The more prominent of these 
lines are designated by the letters B, C, D, E, b, F, G, Hi 
and H2, in order, beginning at the red end of the spectrum. 

Groups of dark lines in the solar spectrum are found to coin- 
cide with groups of bright lines given by iron vapor, sodium 
vapor, hydrogen and other elements, which is an indication that 
relatively cool vapors of these substances exist in the gases which 
surround the sun. Certain dark lines in the solar vSpectrum vary 
in intensity with the altitude of the sun above the horizon. 
These lines are due to the absorption of the earth's atmosphere. 

302. The spectrometer is a spectroscope which is provided with 
a divided circle by means of which the position of the axis of the 
telescope OE in Fig. 380 may be read ofif. The center of the 
circle is beneath the prism P, Fig. 380, the telescope OE is 



DISPERSION. SPECTRUM ANALYSIS. 



427 



carried on an arm which is pivoted at the center of the circle, 
and cross-wires are stretched across the focal plane R V. 

303. The direct-vision spectroscope. — Consider a prism of flint 
glass and a prism of crown glass which give the same deflecdon 
for the middle of the spectrum (by deflection is meant the angle 
a in Fig. 295). Then the flint-glass prism gives a much longer 
spectrum than the crown-glass prism as shown in Fig. 383; and 
the two prisms, if arranged as shown in Fig. 384, disperse a beam 

flint glass 



" II I I II I " ■■ — iia ' 1 11 111 ' 



B C 



D 



E 



B 













i 


B 


C I 


? I 


; 1 


7 G B 


r 



crown glass 



Fig. 383. 

of white light without deflecting the middle portion of the spec- 
trum. 

The ordinary spectroscope would be an awkward instrument 
to use if one were to attempt to hold it in the hands and keep 



crown 



red 




violet 



the collimator CS, Fig. 380, directed towards a source of light 
while looking into the telescope OE. Figure 385 show^s the 
optical parts of a direct-vision spectroscope; S is the slit, CL 



'C\^/C 



CL 



TO 




Fig. 385- 



428 



THE THEORY OF LIGHT. 



is the collimating lens, CFC is a combination of crown-glass 
and flint-glass prisms which does not deflect the middle part 
of the spectrum, but which gives a moderate amount of disper- 
sion, TO is the telescope objective, VR is the spectrum (row 
of images of the slit) , and E is the eye lens. 

304. The achromatic lens. — Consider a prism of flint glass and 
a prism of crown glass which give spectra of the same length. 
Then the crown-glass prism gives a much greater deflection (by 
deflection is meant the angle a in Fig. 295) than the flint-glass 
prism. Two such prisms arranged as shown in Fig. 384 would 
deflect a beam of light without perceptibly dispersing it. Such 
an arrangement might be called an achromatic prism. An 
achromatic lens consists of a converging lens CC of crown glass 
and a diverging lens FF of flint glass arranged as shown in 
Fig. 386 in which WW represents a series of plane weaves of 



W 



+ 




violet 

-h. red 




W 




Fig. 386. 



Fig. 387. 



white light. After passing through the compound lens, waves 
of all v/ave-lengths have approximately the same curvature and 
are focused at the point F' . Short waves (violet light) travel 
slower in both kinds of glass than long waves (red light), but the 
difference between the velocity of red light and the velocity of 
violet light is much greater in flint glass than in crown glass and 
the effect of the double lens of Fig. 386 is to cause the central 
parts of all the waves (of all wave-lengths) to fall behind the 
edge portions by the same amount. 

Let Cr and Ch be the indices of refraction of crown glass 
for red light and for blue light, respectively, and let Fr and Fh 



DISPERSION. SPECTRUM ANALYSIS. 429 

be the indices of refraction of flint glass for red light and for blue 
light, respectively. Consider a converging lens of crown glass C, 
Fig. 387, and a diverging lens of flint glass F. The crown lens 
is assumed to come to a sharp edge and the flint lens is assumed 
to be of zero thickness at the center for the sake of simplicity. 
Let h be the thickness of the crown lens at its center and k 
the thickness of the flint lens at its edge. Consider a plane wave 
of red light passing through the two lenses. The central part 
of the crown lens is equivalent to a distance Crh to be traveled 
in air, and the edge portion of the flint lens is equivalent to a 
distance Frk to be traveled in air. Therefore, the central 
portion of a red wave is retarded with respect to the edge portions 
by an amount which is equal to {Crh — Frk). Consider a plane 
wave of blue light passing through the two lenses. The central 
portion of the crown lens is equivalent to a distance Chh to be 
traveled in air, and the edge portion of the flint lens is equivalent 
to a distance Fbk to be traveled in air. Therefore the central 
portion of the blue wave is retarded with respect to the edge 
portion by an amount which is equal to {Chh — Fbk). If both 
sets of waves, red and blue, are to be focused at the same point, 
the retardation of the central portion with respect to the 
edge portions must be the same for both, that is, 

Crh — Frk = Chh — Fbk 
whence 

K K^r ^ 6 

h ~ Fr — Fh 

This is the necessary relation between h and k to give achro- 
matism. The absolute values of h and k depend upon the 
diameter of the lens and the desired focal length. The curva- 
tures of the various surfaces in so far as they are not fixed by the 
absolute values of h and k and the diameter of the lenses may 
be chosen so as to eliminate spherical aberration.* 

* An example of the complete calculation of an achromatic, aplanatic telescope 
objective, according to Gauss, is given by Otto Lummer in Miiller-Pouillet's Lehr- 
buch der Physik, Vol. II, Part I, pages 573-579. 



430 



THE THEORY OF LIGHT. 



It is impossible to completely compensate the dispersion of a 
converging crown-glass lens by the dispersion of a diverging flint- 
glass lens for reasons which are evident from Fig. 388. This 

flint glass 



B C 



D 



B C 



E F 

] crown glass 



G 







D 



E F 

Fig. 388. 



H 



figure shows spectra of the same total length (Fraunhofer's B- 
line to Fraunhofer's i?-lines) formed by flint-glass and crown- 
glass prisms, and it also shows that the spectra are not of the same 
length as measured between other pairs of lines of the spectrum. 



CHAPTER XXIV. 

INTERFERENCE AND DIFFRACTION* 

305. Reflection with and without phase reversal. — Everyone 
is familiar with crests and hollows in a succession (a train) of 
water waves, and a train of light waves has what we may call 
"crests" and "hollows." Let A A in Figs. 389 and 390 represent 




Fig. 389. 



Fig. 390. 



a train of light waves, full lines being crests and dotted lines 
being hollows. Figure 389 shows a crest c and a hollow h in 
the process of being reflected from the front face of a glass plate, 
and Fig. 390 shows a crest c and a hollow h in the process of 
being reflected from the back face of a glass plate. 

In Fig. 389 the crest c is reflected as a crest c^, and the hollow 
h is reflected as a hollow h\ 

In Fig. 390 the crest c is reflected as a hollow h\ and the 
hollow h is reflected as a crest c\ This conversion of crest to 

* A very good general discussion of interference is to be found in Preston's Theory 
of Light, pages 139-210. An extremely interesting book and one that is easily 
readable is Michelson's Light Waves and their Uses, University of Chicago Press, 
1903. This book is devoted exclusively to the phenomena of interference and to the 
uses of the interferometer. 

A very good general discussion of diffraction is to be found in Preston's Theory of 
Light, pages 211-293. See also Drude's Theory of Optics {translated by Mann and 
Millikan), pages 159-241. 

431 



432 



THE THEORY OF LIGHT. 



hollow and hollow to crest is called phase reversal, and reflection 
from the back of a glass plate is called reflection with reversal 
of phase.* 

306. The colors of thin plates and films. — The most familiar 
example of light interference is in case of the ordinary soap 
bubble the brilliant colors of which are produced by the inter- 
ference of the light which is reflected from the front and back of 
the soap film. Let PP, Fig. 391, represent a thin soap film 
or a thin plate of glass, and let TT represent a train of light waves 
of wave-length X (distance crest to crest or hollow to hollow). 
Let us follow a particular crest cc of the train TT. Part of this 
crest is reflected as a crest from the front of the plate at /; and 
c' is the position of this reflected crest at a given instant. Part 
of the crest passes into the glass, is reflected as a hollow from the 
back of the plate at &, travels through the glass again to f , 
and passes out into the air as a hollow; and h' is the position of 
this hollow at the given instant. Let 2a be the distance traveled 





Fig. 392. 



by h' In the glass. Then while h' is traveling the distance 2a 
in glass and has just reached the point f\ say, the crest c' will 
have traveled // times as far or 2aix in air, so that c' will be 

* There are always two conditions which are associated with each other in a 
wave, namely velocity of medium and distortion of medium-, and one or the other of 
these is necessarily reversed when reilection takes place. This matter is explained 
in simple physical terms in Chapter IX of Franklin and MacNutt's Advanced 
Electricity and Magnetism,. 



INTERFERENCE AND DIFFRACTION. 433 

a distance 2a/i — / ahead of h' , where / is the distance fe. 
The distance 2aix - I is shown as d in Figs. 391 and 392. 

Now the crest c' is associated with a complete train of waves 
which has been reflected from the front of the glass plate; this 
train is represented by T in Fig. 392. Also the hollow h' is 
associated with a complete train of waves which has been re- 
flected from the back of the glass plate; this train of waves is 
represented by T" in Fig. 392. The two wave trains V and 
T" overlap, but to avoid confusion in the diagram they are not 
shown as overlapping. 

If the distance 2a ix — I between c' and h' is an odd number 
of half- wave-lengths (equal to wX/2, where n is an odd number) 
then the two wave trains T' and T" will correspond crest with 
crest and hollow with hollow, and they will merge into a single 
wave train of great intensity. 

If, however, the distance 2aix — / is a whole number of wave- 
lengths (equal to wX/2, where n is an even number) then the 
two wave trains T' and V will correspond crest with hollow, 
and hollow with crest, and they will merge into a single wave 
train of very small intensity, or, indeed, of zero intensity if T' 
and T" are of equal intensities. 

In the first case {2afx — I = n\/2, where n is an odd number) 
the light of wave-length X is reflected in large amount by the thin 
plate as a whole. 

In the second case (2^// — / = n\/2, where n is an even 
number) the light of wave-length X is reflected in small amount 
or not at all by the glass plate as a whole. 

Therefore if white light (containing every wave-length) falls 
on the plate at the particular angle of incidence which is shown 
in Fig. 391, then those particular wave-lengths will be reflected 
in large amounts for which X is contained into 4a /x — 2/ an 
odd number of times, and those particular wave-lengths will be 
but little reflected for which X is contained into ^a/j, — 2I an 
even number of times. 

If the plate PP in Fig. 391 is less than a few ten-thousandths 
29 



_y 



434 THE THEORY OF LIGHT. 

of a centimeter in thickness, then, of all the values of X within 
the limits of the visible spectrum (X = 75 X io~^ centimeter 
for red, to X = 39 X io~^ centimeter for violet) only one or 
two will be contained an odd number of times in ^a^i — 2I; 
therefore such a thin plate reflects, chiefl\^, light of one or two 
wave-lengths and appears brilliantly colored. But if the plate 
is much more than a few ten-thousandths of a centimeter in 
thickness, then many values of X within the limits of the visible 
spectrum will be contained an odd number of times in 4a /-i — 2I; 
therefore such a plate reflects many wave-lengths throughout 
the spectrum and these blend together as approximately white 
light. 

307. The Michelson interferometer is an instrument which 
can be used to measure the wave-length of monochromatic light 
with extreme precision, or, having light of known wave-length, 
it can be used to measure short distances with extreme precision. 

Figure 393 is a general view of a Michelson interferometer, and 
Fig. 394 is a diagram of the same; A and B are two exactly 
similar glass plates, except that B has no silver on it at all 
whereas the face hh of A has a coating of silver so thin that 
about half of a beam of light which reaches the face hh is reflected 
by the silver and about half passes on through the silver; C 
and D are two highly polished silver reflectors. 

Let the lines TT^ Fig. 394, represent the crests of a train of 
plane waves coming towards A A, and let us follow a particular 
crest of this train. Half of this crest is reflected at the back face 
hh as a hollow h and it travels towards D; and half of the 
particular crest goes on through hh towards C as a crest c. The 
reflected hollow h travels to D whence it is reflected back 
to AA, and part of it goes on through A A as the hollow h\ 
Likewise the crest c travels to C whence it is reflected back to 
A A and about half of it is reflected from hh (which is a front face 
with respect to the light returning from C) as the crest c'. 

The hollow h after it is reflected from hh goes once through 
the glass plate A A on its way to D, and it passes again through 



INTERFERENCE AND DIFFRACTION. 



435 



AA on Its return from D. On the other hand the crest c goes 
once through the glass plate BB on its way to C, and it passes 
again through BB on its return from C. Therefore c' and 
h' have passed through the same thickness of glass and if the 
air path from ^ to D is d/2 centimeters shorter than the air 
path from ^ to C (doubled path d centimeters shorter) it is 
evident that h' will be d centimeters ahead of c'. 

Now the hollow h' is associated with a complete train of 
waves T'T', Fig. 395, and the crest c' is associated with a 




Fig- 393- 

complete train of waves T"T" . These two wave trains overlap, 
but to avoid confusion they are not shown as overlapping in Fig. 
395. Therefore, if the retardation d of h^ is an odd number of half 
wave-lengths the trains T' and T" will be crest-on-crest, and 
the result will be an intense beam of light at E, whereas, if d is an 
even number of half wave-lengths the two wave trains T' and 
T" will be crest-on-hollow and the beam at E will be very weak 
(zero intensity if T' and T" are of the same intensity). That 
is to bay, an eye at E in Fig. 394 sees bright light if d is an odd 
number of half-wave-lengths, or no light at all if d is an even 
number of half-wave-lengths. 

The polished silver reflector D is mounted on a sliding carriage 



436 



THE THEORY OF LIGHT. 



and a micrometer screw is arranged to move D towards or 
away from AA. Let the reflector be moved until the eye at 
E sees no light, that is, until the field of vision is dark. Then as 
D is moved the field becomes light and dark alternately, and the 






rp'r - 



k-it^ 



c 




h 



E 



r 






Fig. 394. 



Fig. 395. 



distance I that D is moved is equal to wX/2, where n is the 
number of times the field has become dark during the movement 
and X is the wave-length of the light TT. Thus if / is measured 
(by the micrometer screw) and n is counted we can calculate 
X, or if X is known and n is counted we can calculate the 
value of /. 

In the actual use of the instrument the faces of C and D 
are not exactly at right angles, and the field of vision presents a 
series of bright and dark bands which move sidewise as D is 
moved, and / = wX/2, where n is the number of bright (or dark) 
bands which pass by a given point of the field during the move- 
ment I. 

308. Diffraction. — The spreading of a wave disturbance into 
the region behind an obstacle is called diffraction. This spreading 
action is very prominent in the case of water waves and sound 
waves, but in the case of light waves it requires special arrange- 
ments to make it perceptible. The slight tendency of light 
waves, as compared with the great tendency of sound waves and 



INTERFERENCE AND DIFFRACTION. 



437 





T 












^W 




->• 








water waves to spread into the region behind an obstacle, is due 
to the very short wave-length of light as compared with the long 
wave-lengths of water waves and of sound waves.* 

309. The diffraction grating. ^ — When a train of plane waves 
TT, Fig. 396, strikes an obstacle AB in which there is an 
extremely narrow slit, the portion of the disturbance which passes 
through the slit spreads out, as indicated by the semi-circles 
and divergent arrows in Fig. 396. The 
diffraction grating is an opaque plate in 
which there is a large number of equi- 
distant parallel slits. f Thus, AB, Fig. 
397, represents a diffraction grating, the 
slits of which are numbered o, i, 2, 3, 
4, etc., and TT represents a simple train 
of plane waves, of wave-length X, ap- 
proaching the grating as shown. The 
figure shows the state of affairs after 
twelve successive waves of the train 
TT have struck the grating, that is to 
say, the figure shows twelve successive 

cylindrical waves which have emanated from each slit of the 
grating. According to Huy gens' construction {see Art. 2^1)^ 
every possible surface which is tangent to series of wavelets in Fig. 
SQ'/ is a wave-front. Thus, there is a series of wave-fronts 
parallel to the grating in Fig. 397 at a distance apart equal to X. 
Consider the wavelets which are indicated in Fig. 397 by the 
heavy circles, namely, the second wavelet from slit No. i, the 
fourth wavelet from slit No. 2, the sixth wavelet from slit No. 3, 
the eighth wavelet from slit No. 4, or, in general, the 2wth 

* The theory of diffraction was first developed by Fresnel in 181 5. See Fresnel's 
CEuvres Completes, Vol. I, pages 1-382. A very good discussion of this subject is to 
be found in Edser's Light for Students, pages 427-470, Discussions of diffraction are 
given in Preston's Theory of Light, pages 211-393, and in Drude's Theory of Optics, 
translated by Mann and Millikan, pages 159-241. No attempt is given in this text 
to discuss the theory of diffraction except in so far as the diffraction grating is 
concerned. 

t This description applies to the transmission grating. 



B 

Fig. 396. 



438 



THE THEORY OF LIGHT. 



wavelet from slit No. m. These wavelets have the common 
tangent plane which is indicated by the dotted line W, and there 
is a series of tangent planes W\ W' , W", etc., which are 
parallel to W and at a distance apart equal to X. The existence 
of this series of tangent planes means a train of plane waves 
traveling out from the grating in the direction of the heavy 
arrow in Fig. 397. The value of the angle 6 in Fig. 397 is 

r 




Fig. 397- 



Fig. 398. 



shown more clearly in Fig. 398. Consider the right triangle 
o6p of which the hypothenuse is equal to 6d, where d is the 
distance between centers of adjacent slits. From this triangle, 
we have 



n\ 

sm 9 = —7 
a 



(104) 



INTERFERENCE AND DIFFRACTION. 



439 



t 



in which n is any whole number, X is the wave-length of the 
light which is striking the diffraction grating (see Fig. 397), d 
is the distance between centers of ad- 
jacent slits, and 6 is the angle be- 
tween the plane of the diffraction 
grating and the wave-fronts W, W , 
W", etc., in Fig. 398, or d is the 
angle between the normal to the dif- 
fraction grating and the direction of 
progression of the. wave-train PF, 
W, W", W"', etc., as shown in Figs. 
397 and 398. 

The wave-train which corresponds 
to n = 1, is called the wave- train voi 
of the first order ; the wave-train W, I 
W, W", W"\ etc., in Fig. 397 which 
corresponds to n = 2 is called the 
wave-train of the second order, and so 
on. There are two wave-trains of 
each order as may be seen by a care- 
ful study of Figs. 397, 398 and 399. 

310. The grating spectroscope. — The 
essential features of the grating spec- 
troscope are shown in Fig. 400. A sim- 
ple train of plane waves TT of wave-length X approaches a 
grating AB behind which is a lens LL. The figure shows one 
of the wave- trains of the second order, namely, WWW", etc., 
for which n = 2. This wave train is focused in the focal plane 
PP of the lens at the point F" where the ray r which passes 
through the center of the lens cuts PP. There are two points 
F" corresponding to positive and negative values of 6, and wave- 
trains of the various orders (w = i, n = 2, w = 3, etc.) are 
focused at pairs of points F'F\ F"F", F"'F'", etc. The 
central point F corresponds to n = o. 

The wave-train TT may be thought of as coming from a dis- 




Fig. 399- 



440 



THE THEORY OF LIGHT. 



tant point, such as a star, and the points F, F\ F", etc., are images 
of this star; or the train TT ma}^ come from a nearby point or 




Constructed for « = 2. 
Fig. 400. 



slit 5 and pass through a coUimating lens CC as shown in Fig. 401. 
In this case F, F\ F" , etc., are images of the slit 5. The 
arrangement shown in Fig. 401 is called a grating spectroscope. 




c t\b 




Fig. 401. 



INTERFERENCE AND DIFFRACTION. 44I 

If the incident light TT in Figs. 400 and 401 contains but 
one wave-length, then there will be one image of the slit at each 
of the points F, F\ F' , F" , F" , etc. If, however, the light 
TT contains many wave-lengths, then there will be a group of 
images of the slit near each of the points F\ F' , F" , F" and 
so on, a separate image in each group for each wave-length; and 
each group of images is called a spectrum of the incident light TT. 
The groups of images at F' and F' (for n = \) are called the 
spectra of the first order. The groups of images at F^^ and F'^ 
(for n = 2) are called the spectra of the second order, and so on.* 

* A simple discussion of the concave grating is given on pages 459-463 of Edser's 
Light for Students. The theory of the concave grating is discussed at length by 
H. A. Rowland, American Journal of Science, Series 3, Vol. XXVI, 1883; by R. T. 
Glazebrook, Philosophical Magazine, Series 5, Vol. XVI, 1883; and by J. S. Ames, 
Philosophical Magazine, Series 5, Vol. XXVII, 1892. A very complete discussion 
of the diffraction grating is given by C. Runge in Kayser's Handbuch der Spektro' 
skopie. Vol. I. 



CHAPTER XXV. 

POLARIZATION AND DOUBLE REFRACTION. 

311. Polarization of transverse waves. — When one end of a 
stretched rope is held in the hand and moved rapidly up and down 
or to and fro sidewise a train of waves is sent out along the 
rope from the hand. These waves are called transverse waves 
because, as the waves pass by, a point on the rope moves to and 
fro in a direction at right angles to the direction of travel of the 
waves (the direction of the rope). 

If the hand is moved up and down (in a vertical plane), then 
each particle of the rope w 11 move up and down (in a vertical 
plane) as the waves travel along, and we have what is called a 
plane polarized train of waves. 

If the hand is moved in a small circle around the stretched 
rope as an axis, then each particle of the rope will travel in a 
similar circular path as the waves travel along, and we will 
have what is called a circularly polarized train of waves. 

If the hand is moved in a small ellipse in a plane at right angles 
to the rope, then each particle of the rope will move in a similar 
ellipse as the waves travel along, and we will have what is called 
an elliptically polarized train of waves. 

If the hand is moved sidewise and up and down in an irregular 
way, then an irregular series of waves will travel along the rope. 

Separation of plane polarized waves from irregular waves. — 
A plane polarized series of waves on a rubber tube may be sepa- 
rated out from an irregular series of waves, that is, from a series 
devoid of any persistent and simple type of oscillation, by stretch- 
ing the rubber tube through a narrow slit in a board. Those 
waves of which the oscillations are parallel to the slit, pass through 
the slit freely; those waves of which the oscillations are at right 
angles to the slit, do not pass through the slit at all, but are 

442 



POLARIZATION AND DOUBLE REFRACTION. 443 

turned back or reflected; and those waves of which the oscilla- 
tions are inclined to the direction of the slit are partly transmitted 
and partly reflected. In the latter case the oscillations are 
resolved by the slit into two components, parallel to the slit and 
perpendicular to the slit, respectively; the former component- 
oscillations pass through the slit and the latter component-oscil- 
lations are turned back by the slit, or reflected. The slit thus 
resolves an irregular series of waves into two series of plane 
polarized waves with their planes of oscillation at right angles 
to each other, one series being transmitted and the other series 
being turned back or reflected. 

If the rubber tube is stretched through two slits, the waves 
which are transmitted by the first slit pass freely through the 
second, if the slits are parallel ; the waves are partially transmitted 
by the second slit if the slits are inclined to each other; and the 
waves are completely stopped by the second slit if the slits are at 
right angles to each other. 

312. The optical behavior of tourmaline crystals. Polarized 
light. — A plate of tourmaline, cut parallel to the axis of the 
crystal, transmits only a portion of the light which falls upon it. 
The light which passes through such a plate passes freely through 
a second similar plate when the axes of the plates are parallel, but 
is shut off when the axes of the plates are at right angles. The 
beam of light transmitted by the 
first plate is plane polarized. When 
the second plate is slowly turned 
about the beam of light as an axis, 
the m tensity of the beam which is 

transmitted by the pair of plates changes slowly from maximum 
intensity when the axes of the plates are parallel, to zero inten- 
sity when the axes of the plates are crossed. * This is shown by 
the shading in Fig. 402. 

313. Polarization of light by reflection. — The theory of reflec- 
tion which is discussed in Chapter XIX considers only the 




444 THE THEORY OF LIGHT. 

direction of the reflected ray, whereas a complete theory of reflec- 
tion considers not only the direction of the reflected ray but its 
intensity and the character of its oscillations as to wave-length 
and polarization.* When ordinary light is reflected from the 
polished surface of a transparent substance, such as water or 
glass, the reflected light is partially or totally polarized. The 
degree of polarization varies with the angle of incidence. At 
normal incidence the reflected beam is not polarized at all, as the 
incidence becomes more and more oblique the degree of polariza- 
tion increases, and at a certain obliquity of incidence the polariza- 
tion is complete if the polished surface is perfectly clean. f As 
the incidence becomes still more oblique the degree of polariza- 
tion again falls off. When the reflected beam is completely 
polarized it is at right angles to the refracted beam as shown in 
Fig. 403. The angle of incidence i, Fig. 403, for which the 
polarization of the reflected beam is complete is called the 
polarizing angle. Its tangent is equal to the refractive index of 
the reflecting substance.! Thus for ordinary glass the polarizing 
angle is about 57° as shown in Fig. 404. 

The oscillations of a beam of light which has been polarized 
by reflection are parallel to the reflecting surface. Thus, the 
oscillations of the polarized beams R and R, in Figs. 403 and 
404, are perpendicular to the plane of the paper. § 

314. Reflection of plane polarized light from a glass plate. — 

Consider a beam of plane polarized light R, Fig. 405, which 
falls upon the glass plate A at the polarizing angle. If the 
glass plate A is turned about R as an axis, keeping the angle 

* To give even a bare outline of the fundamental ideas which are involved in the 
complete physical theory of reflection is beyond the scope of this text. This matter 
is discussed in a very satisfactory manner in Drude's Theory of Optics, translated by 
Mann & Millikan, pages 259-302. 

t See Rayleigh, Philosophical Magazine, Vol. 16. pages 444-449, September, 1908. 

t That is, i = tan~i fi. This relation is known as Brewster's law. It may be 
easily shown from Fig. 403, remembering that the ratio of the sines of the angles * 
and r is equal to the index of refraction of the reflecting substance. 

§ Direction of electric field is here referred to. Light waves are electromagnetic 
waves. 



POLARIZATION AND DOUBLE REFRACTION. 



445 



of incidence constant, the amount of light which is reflected varies 
from a maximum when the direction of oscillation of the incident 
beam is parallel to the surface of the glass plate A , to zero when 

I 



.'-'^ 



5r\ 





Fig. 403. 



Fig. 404. 



the glass plate is turned one quarter of a revolution from this 
position. Figure 405 shows the two positions a and c of the 
glass plate A for which it reflects a maximum amount of the 
incident polarized beam R, and the two positions b and d for 
which it does not reflect any of the incident polarized beam R. 




446 THE THEORY OF LIGHT. 

A small portion of the incident beam R is transmitted by the 
plate A in positions a and c, and all of the incident beam R 
is transmitted by A in the positions b and d. By using a 
number of glass plates instead of the single glass plate A in 
Fig. 405, the transmitted beam T^ can be reduced sensibly to 
zero in positions a and c, Fig. 405. This arrangement is 
shown in Fig. 406. 

An interesting experiment which shows the polarization of 
light by reflection is the following: Looking at a varnished pic- 
ture so that light froin a window is reflected from the varnish 
surface to the eye, the details of the picture are invisible because 
of the sheen due to the regularly reflected light. Looking 
through a tourmaline plate (or Nicol prism) which is held in the 
proper position, however, the regularly reflected light is cut off 
because it is polarized, and the details of the picture become 
distinctly visible. 

315. Double refraction. — The theory of refraction which is 
discussed in Chapter XIX is limited to isotropic substances, and 
it considers only the direction of the refracted ray, whereas a 
complete theory of refraction considers not only the direction of 
the refracted ray but its intensity and the character of its oscilla- 
tions as to wave-length and polarization.* To give even a bare 
outline of the fundamental ideas which are involved in the com- 
plete physical theory of refraction is beyond the scope of this 
text, but that property of crystals which is known as double 
refraction may, however, be adequately described. 

Many crystalline substances divide a beam of homogeneous 
light (one wave-length) into two beams by refraction. This 
phenomenon is called double refraction. The crystalline mineral, 

* This part of the physical theory of refraction is closely associated with the 
theory of reflection which is discussed in Drude's Theory of Optics. See footnote 
to Art. 313. A very good discussion of the physical theory of refraction in crystal- 
line substances is given in Drude's Theory of Optics, translated by Mann Sc Millikan, 
pages 308-357. 

Huygens' theory of double refraction which is briefly discussed in this chapter is 
largely geometrical. For full discussion see Preston's Theory of Light, pages 324- 
347. 



POLARIZATION AND DOUBLE REFRACTION 



447 



Iceland spar or calcite, separates the two refracted beams widely 
and therefore shows the effect very distinctl3^ 

In order to get a clear idea of double refracdon, let us consider 
a ray of light e, Fig. 407, which falls upon a glass plate AB 
and is refracted to the point p. If the point p were a luminous 
point, then pe would be a ray passing out from p, and an eye 






f\ 



^ >->>>>i'"i//i',)„),))n,7i^,),,i,i»;iM;j»>>}> 



9 



77^ 



B 



Fig. 406. 



Glass plate. 
Fig. 407. 



placed at e would see the point p at q. Under these condi- 
tions, the plate AB may be turned about the line fg as an 
axis without producing apparent motion of the point q, the 
point p being stationary. 

A beam of common light C, Fig. 408, falling upon a plate 
AB of Iceland spar, becomes two beams and X in the 
crystal. Conversely, a luminous point p, Fig. 408, sends out 
two particular rays o and x, parallel respectively to and 
X, which become parallel rays r and / in the air, so chat an 
eye placed at e would see two images of the point p at q and 
g', respectively. If the plate of spar AB be turned about the 



448 



THE THEORY OF LIGHT. 



line fg as an axis, while the point p remains stationary, then 
one of the images g remains stationary as if AB were a plate 
of glass, and the other image g' moves round g in a small 
circular path. The refracted ray o, or 0, in the spar which 
corresponds to the stationary image q is called the ordinary ray 
inasmuch as it is refracted in the ordinary way as in glass; and 
the ray x, or X, in the crystal which corresponds to the mov- 
ing- image q' is called the extraordinary ray inasmuch as it is 




Fig. 408. 
Plate of Iceland spar. 

not refracted in the ordinary way as in glass. Some crystals 
(bi-axial crystals) divide a beam of common light into two beams 
neither of which follows the ordinary laws of refraction. 

The rays r and /, Fig. 408, are completely polarized, and 
their planes of oscillation are at right angles. This may be shown 
by holding a tourmaline plate (or a Nicol prism) before the eye 
at e. As the tourmaline plate is turned, one after the other of 
the images g and q^ becomes invisible. 

Axis of symmetry of Iceland spar. Optic axis. — Any body 
may be turned one whole revolution about any axis and be in 
exactly its initial position so that to the eye it is just the same as 
if the body had not been turned at all. Consider, on the other 
hand, a symmetrical body like a cube; there are certain axes 
about which a cube may be turned through one quarter, two 
quarters, or three quarters of a revolution and yet appear to the 
eye to be in exactly the same position as at first. This is an ex- 
ample of the kind of symmetry which is exhibited by crystals, 



POLARIZATION AND DOUBLE REFRACTION. 449 

and such an axis is called an axis of symmetry. A crystal of 
Iceland spar has one axis of S3rmmetry about which it can be 
turned one third, or two thirds or three thirds of a revolution 
and appear as if it had not been turned at all.* Not only is the 
form of a crystal of Iceland spar symmetrical with respect to this 
axis, but the physical properties of the substance are also sym- 
metrical with respect thereto. Thas, for example, the heat con- 
ductivity of Iceland spar is different in different directions but it is 
the same in all directions which are equally inclined to the axis 
of symmetry of the crystal. This axis of symmetry of the crystal 
is sometimes called the optic axis of the crystal. 

316. Application of Huygens' principle to double refraction in 
Iceland spar. — The phenomena of double refraction in Iceland 
spar were fully analyzed by Huygens, the discoverer of polarized 
light. Huygens assumed two secondary wavelets to enter a 
plate of Iceland spar from each point on its surface when an 
incident wave reaches that point; one of these wavelets being a 
sphere and the other an ellipsoid of revolution. The envelope 
of the spherical wavelets determines the ordinary refracted wave 
as explained in Art. 256, and the envelope of the ellipsoidal wave- 
lets determines the extraordinary refracted wave. 

The spherical and ellipsoidal wavelets which enter a plate of 
Iceland spar at a point on its surface when an incident wave 
reaches that point, are most easily described by imagining a 
center of disturbance inside of the spar so that the wavelets may 
be a complete sphere and a complete ellipsoid, respectively. Let 
p, Fig. 409, be a center of disturbance in a piece of Iceland spar, 
and let AB be the axis of symmetry of the crystal (optic axis). 
The circle represents the spherical wave which passes out from 
p, and the ellipse represents the ellipsoidal wave. The complete 
ellipsoid is generated by the rotation of the ellipse about AB as 
an axis, and the spherical and ellipsoidal wave surfaces touch each 
other at the poies A and B. 

* The student sho 'Id have access to a paste-board tr.o<lel of a crystal of Iceland 
spar in order to be ab e to understand this statement. 

30 



450 THE THEORY OF LIGHT. 

Straight lines drawn outwards from p in Fig. 409 are called 
rays, The disturbance which constitutes the spherical wave and 
the disturbance which constitutes the ellipsoidal wave may both 
be thought of as traveling outwards along these rays. The 
spherical wave is at each point perpendicular to the rays along 
which it is traveling; but the ellipsoidal wave is not at right 
angles to the rays at every point. 

The velocity of the spherical wave in Iceland spar is the same in 
all directions, namely, 1/1.658 as great as the velocity of light in 
air. The velocity of the ellipsoidal wave in the direction of the 
axis ^-B is the same as the velocity of the spherical wave, and in 
all directions at right angles to AB the velocity of the elUpsoIdal 
wave is 1/1.486 as great as the velocity of light in air. This is 
usually expressed by saying that the ordinary index of refraction 
of Iceland spar Is 1.658, and that the extraordinary index of 
refraction of Iceland spar Is 1.486. The axis AB in Fig. 409, 
or any line parallel to AB, Is the optic axis of the crystal. Any 
plane which includes the optic axis is called a principal plane. 

The oscillations of the spherical wave are everywhere perpen- 
dicular to the principal planes, that is to say, the movement of a 
spherical shell of the ether at the instant that the outwardly 
moving spherical wave reaches it may be Imagined as a momen- 
tary rotatory twitch of the entire shell about AB as an axis, 
followed by a reverse twitch which brings the shell Into its initial 
position. 

The oscillations of the ellipsoidal wave are everywhere in the 
principal planes, that is, the movement of an ellipsoidal shell of 
the ether at the instant that the outwardly moving ellipsoidal 
wave reaches It may be imagined as a momentary twitch which 
carries each part of the shell a short distance towards one pole 
of the ellipsoid along a meridian line and then back again. The 
points A and B are the poles of the ellipsoid in Fig. 409, and 
the meridian lines are the lines of intersection of the principal 
planes with the ellipsoid. 

In Iceland spar, the diameter CD of the ellipsoidal wave is 



POLARIZATION AND DOUBLE REFRACTION. 



451 



greater than its diameter AB parallel to the optic axis, as shown 
in Fig. 409. In quartz, however, the diameter CD of the ellip- 
soidal wave is less than its diameter parallel to the optic axis 
AB , as shown in Fig. 410. 

Figure 411 shows Huygens' construction for the two refracted 
waves in Iceland spar. The figure represents the simple case in 



*^ optic axis 





Fig. 409. 
Wave surfaces in Iceland spar.* 



Fig. 410. 
Wave surfaces in quartz.* 



which the optic axis of the crystal lies in the plane which con- 
tains the incident ray i and which is perpendicular to the re- 
fracting surface, the plane of incidence as it is called. f WW 
represents an advancing wave in air, and WW' is the position 
this wave would have reached at a given instant if it had not 
encountered the piece of crystal. It is required to find the posi- 
tions of the two refracted waves at the given instant. When the 
wave WW reaches the point p, that point is a center of dis- 
turbance, and it sends two wavelets into the crystal, a spher- 
ical^ wavelet and an ellipsoidal wavelet; and at the given instant 
the wavelets from p have had time to travel the distance d in 

* Iceland spar is called a negative crystal and quartz is called a positive crystal in 
view of the difference shown in Figs. 409 and 410. 

t The ordinary refracted ray o always lies in the plane of incidence. In the case 
which is represented in Fig. 411, the extraordinary refracted ray x also lies in the 
plane of incidence, but this is not generally the case. 



452 THE THEORY OF LIGHT. 

air. Therefore the spherical wavelet in the crystal has a radius 
equal to J/ 1.658, and the ellipsoidal wavelet has a minor axis 
equal to the radius of the spherical wavelet and a major axis 
equal to J/1.486. The direction of the optic axis being given, 
both wavelets may be drawn as shown. Wavelets from other 
points in the refracting surface may be determined in a similar 







manner. The envelope of the spherical wavelets is the ordinary 
refracted wave, and the envelope of the ellipsoidal wavelets is the 
extraordinary refracted wave. The ordinary refracted ray is 
the line drawn from p to the point where the spherical wavelet 
from p touches its envelope (ordinary refracted wave). This ray 
is at right angles to the ordinary refracted wave. The extra- 
ordinary refracted ray x is the line drawn from p to the point 
where the ellipsoidal wavelet from p touches its envelope (extra- 
ordinary refracted wave). This ray is generally not at right 
angles to the extraordinary refracted wave. The direction of 
oscillation of the ordinary ray o is determined by the direction of 
oscillation of the spherical wavelets at the points where they 
touch their envelope; this direction is perpendicular to the plane 
of the paper in Fig. 411. The direction of oscillation of the 



POLARIZATION AND DOUBLE REFRACTION. 453 

extraordinary ray x is determined by the direction of oscillation 
of the ellipsoidal wavelets at the points where they touch their 
envelope; this direction is in the plane of the paper in Fig. 411. 
The directions of oscillation of the ordinary and extraordinary 
rays are most easily specified as follows: Imagine a principal 
plane (a plane containing the optic axis) drawn so as to include 
the ray or x m Fig. 411. The oscillations of the ordinary 
ray are at right angles to this principal plane, and the oscillations 
of the extraordinary ray are in this principal plane. 




Fig. 412. 
Optic axis perpendicular to plane of paper. 

Figure 412 shows Huygens' construction for the case in which 
the optic axis is parallel to the refracting surface and at right 
angles to the incident ray i, or, in other words, the optic axis is 
perpendicular to the plane of incidence (the plane of the paper in 
the figure). The significance of the two dotted circles in Fig. 412 
may be understood by referring to Fig. 409. These circles are the 
equatorial sections of the sphere and ellipsoid, respectively. The 
direction of oscillation of the ordinary ray is in the plane of the 
paper and the direction of oscillation of the extraordinary ray is 
at right angles to the plane of the paper in Fig. 412. 

Figure 413 shows Huygens' construction for an incident wave 
WW which is parallel to the refracting surface, the direction of 



454 



THE THEORY OF LIGHT. 



the optic axis being indicated by the short dotted Une. Here the 
ordinary and extraordinary rays coincide, but there are two dis- 
tinct refracted waves nevertheless. 



w w 




^^////^/////j //////// \ /f////// p 

° l\ optic u 

^ // ax.is \\ 
__^-^' / ordinary \ ""-^^^ 


'//////////I '//// 

■ yi 

^' / 


V 

X 


/ wave \ 
_y extraordinaTy ^\^^ 


/ 

f 
y 
X 
^ 


^ 


wave 





\fr 



w/ 



Fig. 413- 



317. The Nicol prism. — A beam of completely polarized light 
(plane polarized) may be obtained by reflection from a glass 
plate as explained in Art. 313. A more convenient arrangement, 
however, for obtaining a beam of completely polarized light is the 
Nicol prism which is constructed as follows: A crystal of Ice- 
land spar is reduced to the form shown in Fig. 414, by splitting 




^'' a! 




end view 

Fig. 414a. 



side view 

Fig. 414&. 



off layers of the crystal.* The line aa in Fig. 414 shows the 
direction of the optic axis. This rhomb of spar is sawed along the 
dotted line AB perpendicular to the plane of the paper in Fig. 
414. The sawed faces and the ends of the rhomb are polished, 

* A crystal of Iceland spar has three cleavage planes, and the faces of the rhomb 
in Fig. 414 are cleavage planes of the crystal. 



POLARIZATION AND DOUBLE REFRACTION. 455 

and the two blocks are cemented together again in their original 
positions with a thin layer of Canada balsam between them. 

A beam of common light C, Fig. 415, is broken up by the prism 
into two beams and x. The ordinary beam is totally re- 



R 




Fig. 415. 

fleeted to one side by the layer of balsam, while the extraordinary 
beam passes on through the prism as shown. Therefore, the light 
R which emerges from the prism is completely polarized, and its 
direction of oscillation is in the plane of the paper in Fig. 415 
(parallel to the shorter diagonal DD of the Nicol prism). 

The reason for the total reflection of the ordinary ray is as 
follows : The index of refraction of the spar for the ordinary ray 
is 1.658 and the index of refraction of the spar for the extra- 
ordinary ray is between 1.486 and 1.658, according to its direc- 
tion, or, say, 1.550. The index of refraction of the balsam is 
between 1.550 and 1.658. That is, the extraordinary ray goes 
from a rare to a dense medium in going from spar into balsam, and 
the ordinary ray would go from a dense to a rare medium in going 
from spar into balsam, the words dense and rare referring to 
large or small index of refraction, respectively. Therefore, the 
ordinary ray may be totally reflected if it strikes the balsam layer 
at sufficiently oblique incidence as explained in Art. 263. On 
the other hand, the extraordinary ray can only be partially 
reflected however oblique its Incidence.* 

318. Action of the Nicol prism on a beam of polarized light. — 
When a beam of ordinary light enters a Nicol prism as shown 
in Fig. 415, the beam Is resolved into two polarized beams o 

* Another device which is sometimes used for completely separating the rays o 
and X in Iceland spar and thereby producing a beam of plane polarized light, is 
Rochon's prism. See Preston's Theory of Light, page 320 



456 



THE THEORY OF LIGHT. 



and x\ the beam x passes through the prism and the beam o 
is reflected to one side. 

A beam of plane polarized light of which the oscillations are 
parallel to the diagonal DD, Figs. 414 and 416, becomes wholly 




Fig. 416. 

the extraordinary beam in the prism, and the whole* of the beam 
passes on through the prism. 

A beam of plane polarized light of which the oscillations are 
parallel to the longer diagonal LL, Fig. 416, becomes wholly 
the ordinary beam in the prism, and it is totally reflected to one 
side by the layer of balsam. 

Consider a beam of plane polarized light of which the oscilla- 
tions are represented in direction and amplitude by the line a in 
Fig. 416. This beam is resolved by the crystalline material of 
the prism into two beams, ordinary and extraordinary, of which 
the amplitudes of oscillation are represented by the lines and 
X, respectively. The extraordinary ray (which passes through 
the prism) has an amplitude x which is equal to a cos 0, and 
the intensity T of the transmitted beam is to the intensity I 
of the incident beam as a^ is to a^ cos^ 0, because the intensity 
of a beam of light is proportional to the square of its amplitude. 

Therefore, 

T = I cos^ 

The intensity T of the beam which Is transmitted by the Nicol 

* Except for a small part which is reflected by the layer of balsam. 



POLARIZATION AND DOUBLE REFRACTION. 



457 



prism varies therefore from a maximum T = I when cos^ ^ 
equals unity, to 2" = zero when cos^ equals zero. 

319. The polariscope. — The polariscope consists of a Nicol 
prism P, Fig. 417, for producing a beam of plane polarized 




Fig. 417. 

light, an arrangement for supporting a crystal plate CC or 
other object to be examined, and a second Nicol prism A through 
which the object CC is viewed. The Nicol prism P is called 
the polarizer and the prism A is called the analyzer. The 
analyzer A is mounted so as to turn freely about the axis of the 
instrument (the heavy dotted line in Fig. 417). 

When the analyzer A, Fig. 417, is turned so that the shorter 
diagonal of its face {DD, Fig. 416) is parallel to the shorter 
diagonal of the face of the polarizer P, then all of the light from 
P passes through A (object CC being removed), and the 
Nicols P and A. are said to be parallel. When the analyzer 
A is turned so that the shorter diagonal of its face is at right 
angles to the shorter diagonal of the face of the polarizer P, then 
no light from P can pass through A (object CC being removed), 
and the Nicols P and A are said to be crossed. 

The arrangement of the polariscope for examining an object 
CC in a convergent beam of polarized light is shown in Fig. 418, 





Fig. 418. 

in w^hich DD is the focal plane of the lens L, and FF is the 
focal plane of the lens L' . Consider the polarized light from P 



458 



THE THEORY OF LIGHT. 




I 1 




U 1 

c 

Fig. 419. 



which passes through any given point g of the plane DD. This 
light passes through the object CC as a pencil of rays parallel 
to the line drawn from g to the center of the lens L, and this 
light is concentrated at the point g' in the focal plane FF. 
The focal plane FF is viewed through an eye lens E and 
through an analyzing Nicol prism A. Under these conditions, 

the brightness and color of any 
point g^ of the focal plane FF 
depends upon the action of the 
object CC on the pencil of rays 
parallel to the line q'c' , and there- 
fore the action of the crystal plate 
upon pencils passing through the 
plate in different directions is indi- 
cated hy the distribution of light and 
color over the focal plane FF, each 
point of the focal plane correspond- 
ing to a definite direction of pencil 
through the crystal plate. Figure 418 shows the two lenses L 
and L' as simple lenses, but in the practical form of instrument 
L and U are compound lenses as shown in Fig. 419. 

320. Rotation of the plane of polarization. The saccharimeter. 

— Many substances, such as crystalline quartz and solutions of 
sugar, cause the direction of oscillation of a beam of polarized 
light to change continuously while the beam is in transit through 
the substance, so that the direction of oscillation of the emergent 
beam is different from the direction of oscillation of the incident 
beam. The angle a through which the direction of oscillation 
of a beam is thus turned (around the beam as an axis) varies 
with the wave-length of the light, and for a given wave-length it is 
proportional to the distance d traversed by the polarized beam 
in the substance; that is, 

a = kd (105) 

in which ^ is a proportionality factor which is called the specific 
rotary power of the substance. 



POLARIZATION AND DOUBLE REFRACTION. 459 

In case of sugar solutions the angle a is proportional to the 
number of grams m of sugar per liter of solution, and to the dis- 
tance d traversed by the polarized beam in the solution; that 

is, 

a = k'md (io6) 

in which k' is a constant. For cane sugar the value of k' is 
0.00665 for sodium light, when a is expressed in degrees, m 
in grams of sugar per liter of solution, and d in centimeters. 
The saccharimeter is a polariscope between the polarizer and 
analyzer of which is placed a long tube with glass ends. A 
solution of sugar is placed in this tube, and the analyzer is 
arranged to permit the measurement of the angle a, whence, 
knowing k^ and d , the strength of the solution may be calcu- 
lated from equation (106).* 

* The saccharimeter of Laurent is described on pages 68-71 of Vol. Ill of 
Frankhn, Crawford and MacNutt's Practical Physics. A very good discussion of 
various forms of saccharimeter is given by Otto Lummer on pages 11 72-1 188 of 
Miiller-Pouillet's Lehrbuch der Physik, Vol. II, part i, Braunschweig, 1897, 



CHAPTER XXVI. 

PHOTOMETRY AND ILLUMINATION. 

321. Radiant heat. Light. — The radiation from a hot body 
may be resolved into simple component parts, each of which is a 
train of ether waves of definite wave-length, and all of these com- 
ponent parts of the total radiation have one common property, 
namely, they generate heat in a body which absorbs them. 
Therefore every portion of the radiation from a hot body is 
properly called radiant heat. The intensity of a beam of radiant 
heat is measured by the heat it delivers per second to an ab- 
sorbing body. Thus, the radiant heat emitted by a standard 
candle represents a flow of about 450 ergs of energy per second 
across one square centimeter of area at a distance of one meter 
from the candle. 

Radiant heat of which the wave-length lies between 39 and 75 
millionths of a centimeter affects the optic nerves and gives rise 
to sensations of light. Therefore radiant heat of which the wave- 
length lies between these limits is called light. These limits, 
which are called the limits of the visible spectrum, are not sharply 
defined, but vary considerably with the intensity of the radiation 
and with the degree of fatigue of the optic nerves, and they 
vary greatly with different persons. 

322. The physical intensity of a beam of light is measured by 
Its perfectly definite thermal effect, that is, by the heat energy it 
delivers per second to an absorbing body. Thus, those parts 
of the radiation of a standard candle which lie within the visible 
spectrum represent the flow of about 9.3 ergs per second across 
an area of one square centimeter at a distance of one meter from 
the candle. Comparing this with the flow of energy which is 
represented by the total radiation from a standard candle (450 
ergs per second across an area of one square centimeter at a 

460 



PHOTOMETRY AND ILLUMINATION. 461 

distance of one meter from the candle), it follows that only about 
two per cent, of the energy radiated by the standard candle lies 
within the visible spectrum, that is, only about two per cent, of 
the radiation from the standard candle is light. Full sunlight 
represents a flow of about two million ergs per second (0.2 of a 
watt) across one square centimeter. About one third or one 
half of this energy is absorbed by the atmosphere. The lumi- 
nous part of the sun's rays represents about four hundred thou- 
sand ergs per second (0.04 of a watt) per square centimeter. 

323. The luminous intensity of a beam of light is presumably 
measured by the intensity of the light sensation it can produce, 
but the intensity of the light sensation which is produced by a 
given beam of light is extremely indefinite. A given beam of 
light entering the eye may produce a strong or weak sensation, 
depending upon various individual peculiarities of the person 
and on the degree of fatigue of the retina; and the vividness of 
the sensation depends upon the extent to which it is enhanced 
by attention. Our sensations are not quantitative in the physical 
meaning of that term; in fact, they enable us merely to dis- 
tinguish objects, to judge whether things are alike or unlike j and 
the certainty and precision with which we can do this is exempli- 
fied in every outward aspect of our daily life. The ratio of the 
luminous intensities of two beams of light is measured by using a 
device to alter, in a known ratio, the physical intensity of one beam 
until it gives, as nearly as one can judge, a degree of illumination 
on a screen which is equal to (like) the illumination produced by 
the other beam. Such a device is called a photometer.* The 
Bunsen photometer is described in Art. 332. 

324. Simple photometry and spectrophotometry. — The meas- 
urement of the light emitted by a lamp is called photometry. 
This measurement is always made by comparing the beam of light 
from a given lamp with the beam of light from a standard lamp 
as explained in Art. 323. 

* A very complete and interesting discussion of Photometric Devices is given 
by C. H. Sharp, on pages 411-506 of Vol. I of the Johns Hopkins University Lec- 
tures on Illuminating Engineering, The Johns Hopkins Press, 191 1. 



462 



THE THEORY OF LIGHT. 



The comparison of the total Hght in a beam from a given lamp 
with the total light in a beam from a standard lamp is called 
simple photometry; whereas the comparison, wave-length by 
wave-length, throughout the spectrum, is called spectrophotom' 
etry.^ 

The fundamental difficulty in simple photometry is that dif- 
ferent lamps usually show differences of color, and these differ- 
ences of color do not disappear when the attempt is made to 
adjust a photometer so that two lamps give equal ilike) illumina- 
tion on a screen. This difficulty is overcome to some extent 
by the use of the flicker photometer.f 

325. Standard lamps. The fundamental light units. — The 

British standard candle is a sperm candle made according to exact 
specifications. t When this candle burns 120 grains of sperm 
per hour it is a standard candle, and the actual candle-power 
during a given test is taken to be a/120 where a is the number 
of grains of sperm actually burned per hour during the test. 

The Hefner lamp,^ so called from its inventor, is a lamp which 
burns pure amyl acetate; the wick and its containing tube are of 
prescribed dimensions, and the wick is turned up to give a flame 
of prescribed height. 

The Vernon-Har court pentane lamp\ is a lamp which burns the 
vapor of pentane. A stream of air flows through a chamber con- 
taining pentane, the air becomes saturated with pentane vapor, 

* A form of spectrophotometer and some typical results of spectrophotometry 
are described on pages 124-125 of Franklin's Electric Lighting. 

t One form of flicker photometer is described on pages 116-117 of Franklin's 
Electric Lighting, The Macmillan Co., New York, 191 2. 

X See American Gas Light Journal, Vol. LX, page 41, 1894. 

§ A full discussion of the Hefner lamp may be found in Fhotometrical Measure- 
ments by Wilbur M. Stine, The Macmillan Co., 1904. In particular, see the dis- 
cussion of Influence of Atmospheric Moisture, Influence of Carbon Dioxide, In- 
fluence of Atmospheric Pressure, and Influence of Atmospheric Temperature on 
the brightness of the Hefner lamp on pages 153-157. 

II The pentane lamp is described on pages 132-134 of Wilbur M. Stine's 
Fhotometrical Measurements. Pentane is one of the more volatile constituents of 
gasolene. 



PHOTOMETRY AND ILLUMINATION. 463 

and this mixture of pentane vapor and air is burned in an Argand* 
burner of prescribed dimensions. 

The Carcel lamp is an Argand burner of prescribed dimensions 
burning rape-seed oil. This lamp has been extensively used as 
a standard lamp in France. 

Light units. — The intensity of the horizontal beam of light 
from a Hefner lamp is called a hefner-unit or a hefner. If a 
lamp were to give one hefner-unit of light intensity in every 
direction, the amount of light, or the so-called flux of light emitted 
by the lamp would be what is called one spherical-hefner. 

The candle, or candle-unit, or candle-power, as it is variously 
called, is a beam of light of which the intensity is i.ii hefner- 
units; that is to say, a horizontal beam from a Hefner lamp has an 
intensity of 0.90 candle-power. This is the definition of the 
' candle-power which is used by the United States Bureau of 
Standards, and the candle-power so defined is called the inter- 
national candle to distinguish it from the old British Standard 
Candle which is now obsolete. 

A lamp which would give one candle- unit of intensity in every 
direction would emit one spherical- candle of light flax. 

For most photometric work nothing is better as a working 
standard than a properly aged and standardized incandescent lamp. 
Such lamps can be obtained from the United States Bureau of 
Standards with certificates specifying their candle-power in a 
prescribed direction when operated with a prescribed voltage 
between their terminals. 

326. Conical intensity and sectional intensity of a beam of 
light. — The expression, intensity of a beam of light, which is 
used in the above definitions of the hefner-unit and candle- 
unit, refers to the amount of light in a unit-sized cone of rays. 
This conical intensity, which it may be called for brevity, is 
expressed in hefners or candles; and it is independent of dis- 

* The Argand burner is a type of burner in which a supply of air is admitted 
to the interior of a flame, as in the familiar student lamp. See article Argand burner 
in any good encyclopaedia. 



464 



THE THEORY OF LIGHT. 



tance, since the light in a given cone of rays always remains in 
that cone.* 

The intensity of a beam of light may also refer to the amount 
of light per unit sectional area of the beam. This sectional inten- 
sity, which it may be called for brevity, decreases as the square 
of the distance from the lamp increases, as explained in Art. 330. 

327. Intrinsic brilliancy of a lamp. — The candle-power of a 
lamp in a given direction divided by the luminous areaf of the 
lamp is called the intrinsic brilliancy of the lamp. 

Examples. — The intrinsic brilliancy of the crater of a powerful 
carbon-arc lamp approaches two hundred thousand candles per 
square inch. The tungsten-filament lamp (i .25 watts per candle) 
has an intrinsic brilliancy of about one thousand candles per 
square inch. The carbon-filament lamp has an intrinsic bril- 
liancy of about three hundred candles per square inch. A kero- 
sene lamp flame has an intrinsic brilliancy of from four to eight 
candles per square inch. 

A lamp of great intrinsic brilliancy is very painful to look at, 
and such a lamp should always be surrounded by a diffusing 
globe or shade so as to hide the luminous surface of the lamp 
itself. Thus, an arc lamp when used indoors is always provided 
with a diffusing globe, and the tungsten lamp should always have 
a diffusing globe or shade when it is used for interior lighting. 

328. Unit of spherical angle. Definition of the lumen. — To 
understand the relationships of the various light units one must 
understand what is called solid or spherical angle. Consider a 
cone, and a sphere with its center at the apex of the cone. Let 
A be the area of the spherical surface which is inside the cone, 
and let r be the radius of the sphere. Then the ratio A/r^ 

* It is assumed in these fundamental definitions that the Hght source is very 
small in size; it requires a very elaborate discussion to establish the fundamental 
ideas of photometry if this assumption is not made. 

A good example of the application of the fundamental ideas of photometry to 
large luminous sources is the paper, "Geometrical Theory of Radiating Surfaces 
with Discussion of Light Tubes," by E. P. Hyde, Bulletin of the Bureau of Standards, 
Vol. Ill, pages 81-104, 1907 

t Projected area at right angles to the given direction. 



PHOTOMETRY AND ILLUMINATION. 465 

measures what is called the spherical angle of the cone.* Thus 
one unit of spherical angle is subtended by one square meter 
of the surface of a sphere of one meter radius, or by one square 
foot of a sphere of one foot radius, and the complete surface of a 
sphere represents 4x units of spherical angle. In the following 
discussion one unit of spherical angle is called a unit-cone. 

Imagine a lamp which gives an intensity of one candle-power 
in every direction. The amount of light, or light flux, passing 
out from such a lamp in one unit-cone is called the lumen of light 
flux. Such a lamp would emit one spherical-candle of light flux, 
inasmuch as the conical intensity is assumed to be one candle- 
power in every direction; but the whole spherical surface repre- 
sents 47r unit-cones, each of which contains one lumen of light; 
therefore there are 47r lumens of light flux in one spherical-candle. 

329. Sectional intensity of a beam of light. Definition of the 
foot-candle. Definition of the lux. — Imagine a lamp which gives 

"Sphere^ 
one foot radi^ 



lunit cone 




one lumeri 
of light 



!pne candle-power 



Fig. 420. 

* To express the value of a spherical angle as the quotient of the spherical area 
divided by square of spherical radius is analogous to the method of expressing a 
plane angle as the quotient of the arc of a circle divided by the radius. 

31 



466 THE THEORY OF LIGHT. 

out one candle-power in every direction, and consider a sphere 
of one foot radius with its center at the lamp. One square foot 
of the surface of this sphere is contained inside of a unit-cone, and 
such a unit-cone contains one lumen of light flux. Therefore 
one lumen of light flux passes through each square foot of the 
surface of the sphere; that is, the light which radiates from the 
given lamp has a sectional intensity of one lumen per square 
foot at a distance of one foot from the lamp. One lumen per 
square foot is sometimes called the foot-candle. That is to say, 
the foot-candle is the sectional intensity of a one-candle-power 
beam at a distance of one foot from the lamp. 

The lux is the sectional intensity of a one-candle-power beam 
at a distance of one meter from the lamp. The lux is one 
lumen per square meter and it is sometimes called the meter- 
candle. 

The relation between the various light units may be kept in 
mind most easily with the help of Fig. 420. 

330. The law of inverse squares. — It is evident that the sec- 
tional intensity of a beam of light from a lamp decreases with 
increasing distance from the lamp. Indeed the sectional in- 
tensity of a beam from a lamp is given by the equation 

C 

^ =J2 (107) 

in which C is the conical intensity of the beam in candle-power, 
and / is the sectional intensity of the beam in lumens per square 
foot at a distance of d feet from the lamp. This is evident when 
we consider that the amount of light in a cone of rays remains 
constant, whereas the sectional area of the cone increases as 
the square of the distance from the apex of the cone. 

Equation (107) expresses what is called the law of inverse 
squares. This law applies strictly to the light which comes from 
a very small portion of the luminous surface of a lamp, a point source 
as it is called. The law is approximately true, however, for 
a whole lamp at distances which are large compared with the 



PHOTOMETRY AND ILLUMINATION. 



467 



size of the luminous surface of the lamp. For example, con- 
sider the light which is given off by a brightly illuminated flat 
circular disk of paper. The sectional intensities of this light at 
points on the axis of the disk at different distances from the disk 
are exhibited in the accompanying table. The heavy-faced num- 
bers show what the sectional intensities would be according 
to the law of inverse squares, and the light-faced figures show 
the actual intensities. 



PAPER DISK 2 FEET IN DIAMETER. 



Distances from disk in feet. 


5 10 


20 40 


80 . 


160 


Sectional intensities according to 

the law of inverse squares 

Actual sectional intensities 


1024 

984.6 


256 

253-5 


64 

63.84 


16 4 

15.99 3.9992 


z 
0.99996 



The error of the law of inverse squares does not exceed two- 
tenths of one per cent, for distances exceeding ten times the maximum 
dimension of the luminous surface of the lamp. 

331. The intensity of illumination of a surface depends upon 
the amount of light falling upon one unit of area of the surface. 
Therefore, intensity of illumination is expressed in terms of 
the same unit as sectional intensity of a beam of light, that is to 
say, intensity of illumination is expressed in lumens per square 
meter (luxes) or in lumens per square foot (foot-candles). 

Example. — The light from a lo-candle-power lamp falls per- 
pendicularly on a sheet of paper at a distance of 2 feet from the 
lamp. Substituting C = 10 and d = 2 in equation (107) 
we find the intensity of illumination / to be 2.5 lumens per 
square foot. 

Oblique illumination of a fiat surface. — When a beam of 
light falls obliquely upon a flat surface as shown in Fig. 421, the 
light is spread over an area greater than the sectional area of 
the beam in the ratio ac/ab, and therefore the intensity of illu- 
mination of the surface is less than the sectional intensity of 
the beam in the ratio ab/ac. That is to say, the intensity of 
illumination of the surface is equal to / cos d where / is the 



468 



THE THEORY OF LIGHT. 



I normal to ac 



sectional intensity of the beam and 6 is the angle shown in 
Fig. 421. 

332. The Buasen photometer. — ^The most extensively used 
device for comparing the conical intensities of the light from two 

lamps is the Bunsen photometer. 
A given lamp and a standard 
lamp are placed at the ends of 
a horizontal bar, and a screen 
(see Fig. 422) of thin paper is 
moved along the bar until the two 
sides of the screen are equally 
illuminated by the two lamps. 
Equal intensities of illumination 
on the two sides of the screen 
indicate equal sectional intensi- 
ties (at the screen) of the beams 
from the two lamps. Let C and C be the conical intensities of 
the beams from the two lamps as shown in Fig. 422. Then the 
sectional intensity at the screen of the beam from the standard 
lamp is Cjd?, and the sectional intensity at the screen of the 




-.i- 



Fig. 421. 



'standard 
\lamp 



screen 



given 
lamp 




beam from the given lamp is C'jd'^, according to equation (107), 
and since these sectional intensities are equal, as above explained, 
we have 



C 



d" 



PHOTOMETRY AND ILLUMINATION. 



469 



whence we get 



c 



m 



(108) 



from which the ratio of the conical intensities may be calculated 
when d and d' have been observed. 

An irregular grease spot on the thin paper screen enables one 
to judge better when the illumination is the same on the two 
sides of the screen. This spot should be made with clean paraf- 
fine, and the excess of paraffine should be drawn out of the screen 
by placing it between folds of absorbent paper and applying a 
hot flat-iron. To facilitate the seeing of both faces of the screen 
simultaneously two mirrors are usually placed as shown in Fig. 

423. 



mirror 



mirror 



light 
from 
'standard 
\lamp 




light 
from 
given 
lamp 



In judging the equality of illumination on the two sides of the 
Bunsen photometer screen, one eye only should be used. In 
using both eyes, one unconsciously looks at one side of the screen 
with one eye and at the other side of the screen with the other 
eye, and the difference between the two eyes leads to a constant 
error of setting. 

With extremely dim lamps (or with bright lamps at great 



470 



THE THEORY OF LIGHT. 



distances from the screen of the Bunsen photometer) the accuracy 
of setting is very low; successive settings taken under such con- 
ditions may deviate from each other by several per cent. With 
increasing intensity of illumination on the screen the accuracy 
of setting increases and reaches a maximum when the intensity 
of illumination on the screen is about one or two lumens per 
square foot.* 

The Bunsen photometer screen, together with the two mirrors 
which are shown in Fig. 423 is usually mounted in a box which is 
called a screen box or sight box. An extensively used substitute 
for the Bunsen sight box is the Lummer-Brodhun sight. box in 
which an optical device (a prism-set) is used for showing portions 
of the two sides of a white opaque screen side by side in the 
same field of view. 

333. Distribution of light around a lamp. — In defining the 



II 6.0- 




Fig. 424. 

* The limit of accuracy of the setting of a Bunsen photometer is about two 
tenths of one per cent., and the error which is introduced by the inaccuracy of 
the law of inverse squares should be considerably less than two tenths of one 
per cent. Therefore the maximum dimension of the luminous surface of a lamp 
should not exceed about one twentieth of the distance of the lamp from the 
photometer screen. See table in Art. 330. 



PHOTOMETRY AND ILLUMINATION. 471 

spherical-candle the idea of uniformity of distribution of light 
around a lamp was introduced for the sake of simplicity. In 
fact, however, no lamp gives complete uniformity of distribution 
of light, but the conical intensity (candle-power) is always greater 
in certain directions and less in other directions. Thus the curve 
in Fig. 424 shows the distribution of light around an ordinary 
carbon-filament lamp without a shade. In this figure the conical 
intensity (candle-power) in each direction is represented to 
scale by the length of the radius vector of the curve. 

334. The illumination of a room.* — A room may be said to be 
well lighted when the eye is easily able to distinguish the various 
objects in the room in minute detail of perception. This com- 
pleteness of visual perception depends upon three conditions: 
namely, (a) a sufficient brightness of illumination, {b) a proper 
location of the light sources so as to bring out that combination 
of soft shadows which is so essential to the perception of form, 
and (c) a proper composition! of the light so as to bring out 
those physical differences in objects which the eye perceives as 
variations of color. 

(a) The necessity of having a sufficient intensity of illumina- 
tion is, of course, known to everyone. The ability to perceive 

* An extremely interesting discussion of the conditions which determine visual 
perception is given by Helmholtz in his popular lecture on The Relation of Optics 
to Painting which is translated (by E. Atkinson) in the second series of Helmholtz's 
Popular Lectures, Longmans, Green & Co., 1903. Everyone who is concerned with 
the practical problems of illumination should read this lecture. Helmholtz's 
Popular Lectures are published in German under the title Vorlrdge und Reden, 
2 volumes, Braunschweig, Vieweg und Sohn, 1884. 

Three lectures in Helmholtz's first series (translated by Dr. Pye-Smith; Long- 
mans, Green & Co., 1873), On the Theory of Vision, also have a bearing upon the 
important practical subject of illumination. 

See also the lectures by Percy W. Cobb and by Robt. M. Yerkes, pages 525-604, 
Vol. IL Johns Hopkins University Lectures on Illuminating Engineering, Baltimore, 
1911. 

The practical method for choosing and locating the lamps for illuminating a 
room is explained on pages 164-170 of Franklin's Electric Lighting, The MacmiUan 
Co., New York, 1912. 

t The composition of light refers to the relative intensities of the various wave- 
lengths which are present in the light. 



472 THE THEORY OF LIGHT. 

fineness of detail (called visual acuity) depends chiefly upon 
intensity of illumination. 

Visual acuity is always measured in an arbitrary way, for example, one may 
measure visual acuity as the distance from one's eye at which clear black print 
of a chosen size may be read, and the dependence of visual acuity upon intensity 
of illumination may be determined by finding the distance at which the given 
type can be read for different intensities of illumination. In this way it is found 
that visual acuity is very low when the intensity of illumination is one or two 
tenths of a lumen per square foot, it increases rapidly up to one or two lumens 
per square foot, and then it increases slowly and reaches a maximum at about 
eight or ten lumens per square foot. 

(b) A room may be sufficiently illuminated by a single arc lamp 
but such illumination is unsatisfactory, even when the eye is 
shaded from the direct light of the lamp, because the excessive 
harshness of the shadows renders the perception of form almost 
impossible. The light from a single brilliant lamp is always 
softened, however, by the reflection from the walls and ceiling 
of a room. 

The second condition is not important where purely flat- 
surface vision is required as in a draughting room, where indeed 
it is important to eliminate all shadows on the sheet of drawing 
paper. 

(c) The importance of the third condition is evident when one 
attempts to distinguish delicate colors by ordinary lamp light. 
Thus the light of an ordinary kerosene lamp is very deficient 
in the short wave-lengths (blue and violet), and a deep blue or 
violet piece of cloth appears almost black by kerosene lamp light. 
False color values are produced in a very striking way by the 
light from a mercury-vapor lamp on account of the almost 
complete absence of the longer wave-lengths (red) in the light 
from this lamp. The most striking illustration of false color 
values, however, may be obtained by illuminating a batch of 
brilliantly colored worsteds by the light from a sodium flame in 
a room from which all white light is excluded. All differences of 
tint disappear under these conditions, and a given piece of 
worsted merely appears to be light or dark according as it is able 
or unable to reflect the yellow light of the sodium flame. 



PHOTOMETRY AND ILLUMINATION. 473 

335. Glare. — The presence of excessively brilliant lamps or 
excessively brilliant patches of Ught in a field of vision greatly 
hinders visual perception. The eye adapts itself automatically 
to the brightest lights in the field of view, and all perception of 
detail in the shadows is lost. This effect is called glare, and it is 
especially marked when the field of vision includes a bright 
unshaded lamp. 

The explanation of glare is as follows: In the first place, a 
beam of light entering the eye from a bright source illuminates 
the whole interior of the eye just as a beam of sunlight entering 
a window illuminates a room. This diffused light in the eye 
illuminates and excites the entire retina, including those portions 
where the images of the deeper shadows fall, and thereby tends 
to obliterate all detail of perception. In the second place the 
portions of the retina upon which the brilliant light falls become 
greatly reduced in sensitiveness by fatigue, the continual wander- 
ing of the eye brings the image of a dark region upon this fatigued 
portion of the retina, and the result is almost total blindness like 
that produced when one looks out of a window and then turns 
towards a dark corner of a room. In the third place the pupils 
of the eyes contract greatly when there is a bright light in the field 
of vision, and this contraction lessens the effective brightness 
not only of the bright portions of the field, but also of the deep 
shadows; but the deep shadows are already insufficiently illu- 
minated and the contraction of the pupils of the eyes tends to 
make them (the shadows) appear like black patches entirely 
devoid of detail. 

It is very important, in arranging for the illumination of a 
room, to place the lamps outside of the field of vision if possible, 
so that no light can enter the eye directly from the lamps and 
render the eye insensible to the delicate shading of surrounding 
objects. The excessive discomfort that is produced by the glare 
of improperly located lamps, such, for example, as the exposed 
footlights of a poorly arranged stage, is due not only to the 
physical pain that is associated with long-continued looking at a 



474 THE THEORY OF LIGHT. 

bright light but more especially to the incessant effort of trying 
to peer into the dark region beyond. 

When a lamp cannot be removed from the field of vision the bad 
effects of glare may be greatly reduced by enlarging the luminous 
surface of the lamp by means of a translucent globe or shade. 

336. Small lamps versus large lamps. — A small lamp, as the 
term is here used, is a lamp which gives a small amount of light; 
and a large lamp is a lamp which gives a large amount of light. 
A given amount of light can be produced more cheaply by large 
lamps than by small lamps because large lamps are, as a rule, 
more efficient (less watts per lumen) than small lamps and be- 
cause a few large lamps are cheaper to install and cheaper to 
maintain than many small lamps. The use of large brilliant 
lamps is, however, limited by two conditions as follows: 

(a) Satisfactory distribution of light. — ^Whenever it is necessary 
to use a large number of lamps in order to get a satisfactory 
distribution of light, small lamps are used because to use large 
lamps would give an unnecessarily large quantity of light. It 
is not desirable, however, to have light too uniformly distributed 
(by using a great number of small lamps) because the resulting 
illumination is flat, that is, devoid of satisfactory shadows. 

{b) Elimination of glare. — A large lamp in one's field of vision 
produces a much more unpleasant glare than a small lamp, and 
therefore (even if a proper distribution could be secured) it is 
not advisable to use large lamps in rooms with low ceilings be- 
cause with a low ceiling a lamp cannot be placed high enough to 
remove it entirely from one's field of vision. Large lamps must 
be placed high overhead. This is especially true of lamps which 
have a high intrinsic brilliancy; such lamps must not be placed 
in the field of vision unless they are shaded. 

When the indirect system* of lighting is employed, however, 
very large brilliant lamps can be used in small rooms. 

*]A good discussion of indirect lighting is given by L. B. Marks, Johns Hopkins 
University Lectures on Illuminating Engineering, Vol. II, pages 691-702, Baltimore, 
191 1. See also an article by J. R. Cravath, Transactions of the Illuminating En' 
gineering Society, Vol. IV, pages 290-306, 1909. 



CHAPTER XXVII. 

COLOR* 

337. Sensations of brightness and sensations of color. — A 

beam of light which falls upon the retina produces two distinct 
sensations, namely, a sensation of brightness and a sensation of 
color. The intensity of the sensation of brightness depends 
upon the physical intensity of the light,! and the character and 
vividness of the sensation of color depends upon the wave-length 
of the light, or when the light is not homogeneous, upon the 
relative intensities of the various wave-lengths which are present 
in the light. 

338. Luminosities of various parts of the spectrum. — The dif- 
ferent parts of the spectrum differ greatly in brightness, and by 




4500 



6000 



ATioli&t 



55(X) 6000 

Fig. 425. 



6600 7000 



Bad 



* The most complete treatise on color from both the physical and physiological 
points of view is to be found in Helmholtz's Handbuch der Physiologischen Optik, 
pages 275-384. 

An extremely interesting discussion of brightness and color effects in their bearing 
upon painting is to be found in Helmholtz's popular lecture on The Relation of 
Optics to Painting, see footnote to Art. 334. 

An interesting book is Ogden Rood's Text Book of Color or Modern Chromatics, 
New York, 1881. 

t Also to some extent on the wave-length, see next article. 

475 



I 



476 THE THEORY OF LIGHT. 

no means in proportion to the energy of the different parts. 
The ordinates of the dotted curve in Fig, 425 represent the energy 
or heating intensities of the various parts of the spectrum of the 
light from a kerosene lamp, and the ordinates of the full-line 
curve represent the degrees of brightness of the various parts 
of the spectrum to the normal eye. The abscissas represent 
wave-lengths.* 

339. Color sensations due to homogeneous light. — Light of 
one wave-length is called homogeneous light as explained in 
Art. 297, and the various wave-lengths of homogeneous light 
produce the familiar series of colors of the spectrum as seen in a 
spectroscope. Newton recognized and named seven colors in the 
spectrum: red, orange, yellow, green, blue, indigo and violet. 
As a matter of fact, however, about 150 steps are made in going 
through the spectrum from one tint to the next which can barely 
be distinguished from it. That is to say, there are about 150 
distinguishable tints in the spectrum. The most vivid colors 
in the spectrum are the extreme red, the green and the blue or 
violet. 

340. Color sensations due to mixed light. — It is evident that 
there is a possibility of an infinite variety of mixtures of the 
various wave-lengths of light, according to the relative intensities 
of the various wave-lengths that are present in the mixture, and 
the number of distinguishable color sensations which may be 
produced by. various light mixtures is, according to Titchener,t 
about 30,000, and, according to Rood,t a much larger number. 

White light. — Sun light, or any light approaching sun light 
in composition, is called white light. The sensation produced 
by such light, aside from complications growing out of contrast 
effects, is called white. 

* A simple method for determining this luminosity curve, due to Tufts, is 
described in Franklin, Crawford and MacNutt's Practical Physics, Vol. Ill, pages 
26-27, The Macmillan Co. 

t Titchener, An Outline of Psychology, page 66. 

X Rood, A Text Book of Color, Chapter IX. 



COLOR. 477 

Saturated and diluted colors. — Mixed light which contains a 
great excess of one wave-length or narrow group of wave-lengths 
generally produces a vivid sensation of color. Such a color is 
sometimes called a saturated color. Mixed light which approaches 
white light in composition, having only a slight excess of one 
wave-length or group of wave-lengths, gives a pale sensation of 
color. Such a color is sometimes called a diluted color. 

341. The cause of color in natural objects. — Colored objects 
occurring in nature owe their color to the fact that they send to 
the eye light of which the composition (relative intensities of the 
different wave-lengths) differs more or less widely from the com- 
position of white light. 

Examples. — (a) Hot gases and vapors give off light which 
differs widely in composition from white light. Thus, most of 
the color effects in fireworks are produced by the use of salts of 
the various metals. These salts are vaporized and the hot vapors 
give off brilliantly colored lights. 

Q)) Many substances reflect certain wave-lengths in great 
excess. Thus powdered ultramarine blue reflects about 20 per 
cent of the blue, indigo and violet, and only about 5 per cent of 
the red and yellow of ordinary day light. 

(c) Many substances transmit certain wave-lengths in great 
excess. Thus an orange colored solution of potassium chromate 
of a particular concentration and thickness, transmits about 80 
per cent of the red and yellow, about 30 per cent of the green, 
and Jess than 2 per cent of the blue, indigo and violet of ordinary 
day light. 

342. Mixing of colors. — It has been known since the earliest 
days of painting that two colors blend completely and give a 
single resultant color or tint when mixed. 

To understand the difference between the mixing of colored 
lights, and the mixing of pigments let us consider the following 
example. 

Let us think of white light as containing 100 parts of red light, 
100 parts of green light, and 100 parts of violet Ught. 



478 ^THE THEORY OF LIGHT. 

Given a pane of yellow glass which transmits one half of the 
red (50 parts), one quarter of the green (25 parts), and not any- 
violet. 

Given a pane of blue glass which transmits not any of the red, 
one quarter of the green (25 parts), and one half of the violet 
(50 parts). 

(a) Imagine the two panes to be set side by side in a window 
frame, then the light which comes into the room through the red 
pane would be mixed with or added to the light which comes into 
the room through the blue pane, and we would have in the room 
50 parts red light, 50 parts green light, and 50 parts violet light, 
which would be white light, somewhat weakened, of course. 

(6) Imagine the two panes to be set one over the other as 
one double-thickness pane. Then the two panes together would 
transmit no red light and no blue light, but they would transmit 
I of i or ^^g of the green which would be 6 J parts of green light. 
That is to say, we would have green light in the room. 

(a) Yellow and blue lights when mixed (added) give white. 
{h) Yellow and blue glasses laid over each other give green. 
Something similar to {h) takes place when pigments are mixed. 
Thus a yellow pigment mixed with a blue pigment gives a green 
pigment. 

The simplest device for mixing two colored lights is that which 

is shown in Fig. 426, in which 
a is a plate of unsilvered glass, 
and h and c are bits of .col- 
ored paper or colored paint. 
The light from h passes through 
the glass plate a into the 
eye, and the light from c is 
reflected by the glass plate into 




d 



p. ^^ ' the eye, as indicated by the 

dotted lines. 
The most convenient arrangement, perhaps, for mixing colored 
lights is the color top. This consists of a rotating spindle upon 



COLOR. 479 

which are mounted disks of colored paper which are sHtted in 
such a way that any desired sector of the face of each disk may 
be exposed to view. When this composite disk is rotated rapidly 
the colors blend and give a single sensation of color, the tint of 
which may be modified at will by varying the amounts of the 
respective disks that are exposed. 

Complementary colors. — Two colors (colored lights) which 
produce white light when mixed (added) are called complementary 
colors. Thus yellow and indigo-blue are complementary colors. 

343. Matching of colors by mixtures. Two-color vision and 
three-color vision.' — Any color may he matched hy a mixture^ in 
proper proportions, of a saturated red light, a saturated green light, 
and a saturated violet light. This is an experimental fact which 
was discovered by Thomas Young. 

For some persons {red blind) any color may he matched hy a 
mixture, in proper proportions, of a saturated green light and a 
saturated violet light. For other persons {green blind) any color 
may he matched hy a mixture, in proper proportions, of a saturated 
red light and a saturated violet light. Persons for whom any 
color may be matched by mixing two saturated colors are said to 
have dichroic or two-color vision. Persons for whom the matching 
of any color requires the mixing of three saturated colors are 
said to have trichoic or three-color vision. Dichroic vision is 
commonly called color blindness. About four per cent, of the 
male population and about four tenths of one per cent, of the 
female population of the civilized world are color blind. 

344. The Young-Helmholtz theory of color. — The fact that any 
color can be matched by a proper mixture of three saturated 
colors led Thomas Young, in 1801, to infer the existence of three 
primary color sensations. Helmholtz attributed each of these 
primary sensations to a distinct set of nerves in the retina of the 
eye. The nerves which upon excitation give the primary sensa- 
tion of red are called the red nerves, those which upon excitation 
give the primary sensation of green are called the green nerves, 
and those which upon excitation give the primary sensation of 



48o 



THE THEORY OF LIGHT. 



violet are called the violet nerves. Simultaneous excitation of all 
three sets of nerves gives a blended sensation the character of 
which depends upon the relative intensities of excitation of the 
respective sets of nerves. 

A person having dichroic vision has, according to the Young- 
Helmholtz theory, only two sets of color nerves, and therefore 
only two primary color sensations. Persons who do not have 
the primary sensation of red are said to be red blind; and persons 
who do not have the primary sensation of green are said to be 
green blind. No clearly defined case of violet blindness has ever 
been found. 

The Young-Helmholtz theory of color is not generally accepted. 
Physiologists are inclined to reject it for lack of microscopical evi- 
dence of the existence of the three sets of nerves, and psycholo- 
gists are inclined to reject it mainly because of the difficulty of 
explaining the great number of distinguishable color sensations 
by the varying intensities of sensation of three sets of nerves. 
The theory gives, however, a very clear representation of the 
experimental facts of color mixing and very satisfactory explana- 
tions of contrast effects and color blindness. 

Sensitiveness of the color nerves to lights of different wave- 
length. — When a given sensory nerve is excited by different 

means the sensation is 
always the same as ex- 
plained in Art. 239. 
Thus the excitation of 
the red, green or violet 
nerves always gives sen- 
sations of red, green or 
violet, respectively, how- 
ever the excitation may 
be produced. The or- 
dinates of the curves in 
Fig. 427 represent the relative degrees of sensitiveness of the 
respective sets of color nerves to different wave-lengths of light. 




Fig. 427. 



COLOR. 481 

These curves are from measurements made by Koenig. As these 
curves show, every wave-length in the spectrum can excite the 
red nerves and in some degree give the sensation of red. The 
green nerves can be excited more or less by any wave-length 
between Fraunhofer's C line and G line; and the violet nerves 
can be excited more or less by any wave-length between the E 
line and the H line. 

Any particular wave-length of light excites all three sets of 
nerves more or less, and produces a blended sensation. 

345. Contrast effects. — Two complementary colors placed side 
by side tend to intensify each other as the eye glances from the 
one to the other. Any two colors placed side by side, if they are 
at all different, tend to become complementary in appearance 
as the eye glances from one to the other. This contrast effect, 
as it is called, may be shown very strikingly by placing two small 
pieces of pale green paper exactly alike, one on a large sheet 
of red paper and the other on a large sheet of blue paper. The 
two pieces of green paper are so greatly changed by the opposite 
contrasts that one can scarcely believe that they are physically 
alike. A bit of white or grayish paper placed upon a broad 
sheet of brilliantly colored paper assumes a very distinct hue 
complementary to the surrounding color. 

The explanation of contrast effects in terms of the Young- 
Helmholtz theory of color is as follows: White light affects all 
three sets of color nerveg in certain relative proportions. A 
colored light also affects all three sets of nerves, but the vividness 
of the color sensation which is produced depends upon the pre- 
ponderating excitation of one or two sets of nerves. When one 
looks at a brilliant color, one or two sets of color nerves become 
more or less fatigued, and, under these conditions, a neutral tint 
produces a decreased excitation of these fatigued nerves or a rela- 
tively greater excitation of the unfatigued set of nerves, thus tend- 
ing to produce the complementary color sensation. 

346. Color blindness. — Color-blind persons show marked pe- 
culiarities in their sensations of brightness and color. Thus the 

32 



482 



THE THEORY OF LIGHT. 



ordinates of the dotted curve in Fig. 428 represent the relative 
degrees of brightness of the various parts of the spectrum of 
gas Hght as seen by the normal eye (the dotted curve in Fig. 428 
is the same thing as the full-line curve in Fig. 425), and the 
ordinates of the full-line curve in Fig. 428 represent the relative 



80 


- 




/ 
/ 
1 
1 

1 
1 
1 ^ 


\ 
\ 
\ 

\ 
\ 
\ 


60 


V 

- t 
3 












z 




1/ 
// 


1 \ 




S 




1/ 


40 


3 
- J 














n > 










C > 






Jt 




z \ 






/ 




=> \ 


20 


^ 


/ 


Y//KVE LENGTH 

1 


V V, 



BLUE 



5000 6000 

Fig. 428. 



7C00 

RED 



degrees of brightness of the various parts of the spectrum of 
gas light to a red-blind person.* 

Color sensations due to homogeneous light. — Figure 429 shows 
roughly the appearance of the spectrum to persons with normal 
trichroic vision, to green-blind persons, and to red-blind persons 







G 




r E D 


c 


B 


TP.ICHROfe 


















VIOLET 




BLUE 


GREEN 


VELLOW 


ORANGE 




RED 


bREEN-BLIND 




















VIOLET 






WHITE 






RED 






RED-BLIND 



















VIOLET 



WHITE 



Fig. 429. 



respectively. The red and violet nerves of a green-blind person 

are both excited more or less by any wave-length of light, and 

* From measurements made by Ferry, American Journal of Science, Vol. 44, 1892. 



COLOR. 483 

that particular wave-length which excites these two sets of nerves 
in the same way that ordinary white light excites them gives to 
the green-blind person the sensation which he calls white. This 
wave-length lies between the E and F lines of Fraunhofer 
according to Fig. 427 and as indicated in Fig. 429. 

Similarly the red-blind person sees what he calls a "white" 
region in the middle part of the spectrum. 

Color sensations due to mixed light. — Color-blind persons show 
marked peculiarities of color sensations due to mixed lights. 
These peculiarities are very complicated, in fact, they are hardly 
the same for any two persons. The following article which de- 
scribes the Holmgren test for color blindness, gives some idea 
of the peculiar color sensations of color-blind persons. 

347. The Holmgren test for color blindness. — All systems of 
signalling upon railways and at sea, depend more or less upon 
the recognition of colored lights, and consequently the character 
of the color sense of employes is a matter of vital importance. 
It is fortunately possible to detect dichroic vision with certainty 
even when the existence of the peculiarity is unsuspected by the 
subject, or when the subject attempts to conceal the matter. The 
simplest method of performing such tests was inv^ented by Holm- 
gren, of Upsala, after the occurrence (in Sweden) of a dreadful 
railway accident due to the color blindness of an employe. The 
Holmgren apparatus consists simply of a collection of colored 
worsteds which includes a large variety of colors of every degree 
of saturation and a number of neutral grays of various degrees 
of brightness. There are also three larger skeins which are 
called the confusion samples. One of these is a pale green, cor- 
responding very nearly in tint (but of course not in saturation) to 
the portion of the spectrum which appears white to both red- 
blind and green-blind persons, as described in Art. 346. An- 
other confusion sample is pale magneta, which is a blend of red 
and violet with much white, and the third confusion sample is a 
brilliant red. 

Both red-blind and green-blind persons invariably select a 



484 THE THEORY OF LIGHT. 

variety of neutral or gray worsteds as corresponding most closely 
with the pale green confusion sample. 

Red-blind persons inv^ariably select violets and blues as corre- 
sponding most closely with the magenta confusion sample. 

Both red-blind and green-blind persons are inclined to select 
brilliant greens as well as brilliant reds in matching the bright 
red confusion sample. 



PART V. 
THE THEORY OF SOUND. 

A most interesting book is Sound by John Tyndall, published by Longmans, 
Green & Co., in 1875. This book is almost entirely descriptive in character. 
One of the best simple treatises on sound is Poynting and Thomson's Sound, pub- 
lished by Charles Griffin and Co., in 1899. This book contains some of the more 
important discussions of the mathematical theory of sound. The most complete 
treatise on the theory of sound is Lord Rayleigh's Theory of Sound in two volumes. 
The second edition of this work was published by Macmillan and Company in 
1894. The most complete treatise on sound from the physiological point of view 
is Helmholtz's Tonempfindungen. (Translated by Alexander J. Ellis, Longmans Sc 
Co.) This book gives, in particular, a very* exhaustive discussion of the physical 
theory of music. Professor D. C. Miller's The Science of Musical Sounds, The Mac- 
millan Co., New York, 1916, is an important contribution to the theory of music. 

Professor W. C. Sabine's important researches on Architectural Acoustics are 
described in a very simple and practical way in Franklin and MacNutt's Light 
and Sound, pages 282-290, The Macmillan Co., New York, 1909. 



CHAPTER XXVIII. 

TONES AND NOISES. LOUDNESS, PITCH AND QUALITY. 

348. The method of optics and the method of acoustics. — In 

the study of light we have been concerned chiefly with the phe- 
nomena which are associated with transmission, such as the 
phenomena of reflection, refraction and diffraction. In the study 
of sound, on the other hand, we are concerned chiefly with the 
mode of generation of sound waves and with the effects which are 
produced by sound waves when they strike a body. The phe- 
nomena of reflection, refraction and diffraction are as real in the 
case of sound waves as in the case of light waves but these 
phenomena of sound are not of great importance. 

The more advanced study* of optics is concerned with the 
mode of generation of light waves in luminous bodies, and with 
the effects of light waves upon bodies or substances upon which 
light waves fall, but these things are beyond the scope of this 
text. 

349. Noises and musical tones. — Sound, in the physical sense 
of the word, consists of waves in the air which travel from the 
sound producing body to the ear, as stated in Arts. 241 and 242; 
these waves are produced by quick movements of the sounding 
body, and when they fall on the ear they produce the sensation 
of sound. 

When a sounding body performs a succession of similar move- 
ments to and fro, its motion is said to be periodic, and it produces 
a succession of similar sound waves, or a wave-train. The 
sensation corresponding to such a wave- train is called a tone. 

* The branches of optics here referred to are spectrum analysis including much 
recent work on the absorption and emission of X-rays. Also the theory of refrac- 
tion, including the theory of dispersion, has to do with the character of the dis- 
turbance produced in a body which receives a beam of light. The theory of dis- 
persion is discussed in Drude's Theory of Optics, pages 382-389. 

487 



488 THE THEORY OF SOUND. 

Sound sensations which cannot be classified as tones or com- 
binations of tones are called noises. For example, rattling noises 
are due to irregular successions of sharp clicks, each of which 
sends a single wave pulse to the ear. Hissing and roaring noises 
are due to complex and rapidly varying combinations of tones. 
In the case of hissing noises the tones, which may be few in num- 
ber, are of very high pitch,* and in the case of roaring noises 
the tones are usually numerous and of lower pitch. 

Musical tones are generally accompanied by characteristic 
noises. Thus, the whispering noises of the breath and the sounds 
of the consonants used in articulation accompany the musical 
tones of a singer; and the faint noises produced by the fingers, 
keys and pedals always accompany piano music. Many noises, 
on the other hand, are accompanied by distinctly audible musical 
tones; thus, a light hammer-blow upon a floor or upon a piece 
of furniture produces a musical tone of short duration which is 
often easily distinguishable. 

350. Loudness of tones. — It is a familiar fact that the sound 
emitted by a vibrating body, such as a guitar string, increases in 
loudness with increase of the amplitude of the vibrations (by 
amplitude is meant half the distance through which the middle 
of the string swings to and fro). When sound waves travel 
through the air, any given particle of the air oscillates to and fro 
through a certain amplitude, and the amplitude of the air oscilla- 
tions increases with the amplitude of the given vibrating body 
which produces the sound waves. f The loudness of a tone depends 
upon the amplitude of the air oscillations, 

351. Pitch of tones. — Consider a musical instrument such as 
the piano or harp. The short strings have a greater frequency of 
vibration than the longer strings, and that quality of a musical 
tone which depends upon the frequency of the vibrations is called 

* See Art. 351. 

t A vibrating string may oscillate through a wide amplitude and emit a much 
weaker sound than is emitted by a board or plate which vibrates through a much 
smaller amplitude, so that the amplitude of the vibrating body alone does not deter- 
mine the amplitude of the air oscillations. 



TONES AND NOISES. 489 

pitch. The pitch of a tone is high or low according as the vibra- 
tion frequency is great or small. Two vibrating bodies which 
give tones of the same pitch are said to vibrate in unison. 

Determination of pitch.* — The direct determination of pitch is 
accomplished by counting the number of vibrations in a given 
time. The siren is sometimes used for this purpose. It consists 
of a circular metal disk mounted on a shaft which is geared to a 
revolution counter. The disk has one or more circular rows of 
equidistant holes, it rotates near to the wall of a chamber contain- 
ing air under pressure, and the holes in the disk come before 
apertures in the wall in rapid succession. The puffs of air thus 
produced blend into a tone, the pitch (number of vibrations per 
second) of which is known from the observed speed of the disk 
and the known number of holes in the row. The disk is some- 
times driven by an electric motor and sometimes by the action of 
the puffs of air. In the latter case the holes in the disk are in- 
clined like the vanes of a wind-mill. 

Standards of pitch. — A vibrating body which has an invariable 
frequency of vibration may have its pitch accurately determined 
once for all, and the pitch of any sound may then be determined 
by comparison with this standard. Standards of pitch are 
usually in the form of tuning forks (see Art. 365). 

Pitch limits of audibility. — When a vibrating body has a 
frequency less than about 30 complete vibrations per second it 
does not produce a sensation of tone, but the ear perceives the 
separate impulses as a fluttering noise. When a vibrating body 
has a frequency greater than about forty thousand complete 
vibrations per second it does not produce any sensation of sound 
at all. 

352. Tone quality or timbre. — It is a familiar fact that one can 
easily distinguish musical tones produced by different musical 
instruments independently of differences in loudness and pitch. 
Thus, a note of the same pitch and loudness produced by a singer 
and by a violin are so entirely different that there is no difficulty 

* See Poynting and Thomson's Sound, pages 36-47. 



490 



THE THEORY OF SOUND. 



whatever in distinguishing the one from the other. This differ- 
ence in quaHty depends upon the character of the oscillations of 
the vibrating body (or upon the character of the oscillations of the 




ynoo ct 



Fig. 430. 

air particles in the sound waves which are produced by the 
vibrating body). Thus, a point on a violin string usually moves 
slowly in one direction and more rapidly in the opposite direction, 

like the up and down motion 
which would have to be given to 
a pencil point to trace the zig- 
zag line in Fig. 430 with the 
paper moving uniformly to the 
left. 

Figure 431 shows the character 
of the vibrations which corre- 
spond to the various vowels when 
sung by a baritone voice. It 
must not be thought from this 
figure, however, that a given 
vowel sound is prduced by a 
characteristic type of oscillation. 
This matter is quite fully dis- 
cussed in Art. 371. 

353. Simple and compound vibrations. — When a particle 
moves to and fro along a straight line performing simple har- 
monic motion, its vibrations are called simple vibrations. When 
the to and fro motion of a particle is periodic but not simply 
harmonic, its vibrations are called compound vibrations. 

Examples. — The vibrations of a pendulum bob, and the 
vibrations of the prongs of a tuning fork are simple vibrations. 




n^At 



fit 



I?'}^ 



zu 



Fig. 431- 



TONES AND NOISES. 



491 



The vibrations of a reed which is slowly lifted and quickly 
dropped by the successive cogs of a rotating wheel would be 
compound vibrations. 

Graphical representation of simple and compound vibrations. 
— Imagine a pencil point p in Fig. 432 to vibrate up and down 

'.A 




Fig. 432. 

along the line AB, and imagine the paper to move at uniform 
velocity to the right. Then the point p will trace a curve. If 
the vibrations of p are simple the curve cc will be a curve of 
sines. If the vibrations of p are compound the curve cc will 
be a periodic curve, that is, each section of the curve will be 
exactly like every other section, but it will not be a curve of sines. 

A 
P 




reference 



axis. 




Fig. 433. 

The curve in Fig. 432 is a curve of sines and it represents 
simple vibrations. The curve in Fig. 433 is a periodic curve 
(not a curve of sines), and it represents compound vibrations. 
The curve in Fig. 430 and all but the first curve in Fig. 431 
represent compound vibrations. 

Definitions. — The number of complete vibrations (round-trip 
vibrations) per second is called the frequency of the vibrations. 
The duration of one complete vibration is called the period of the 



492 THE THEORY OF SOUND. 

vibrations. One half of the distance through which the vibrat- 
ing particle moves to and fro is called the amplitude of the 
vibrations. 

354. Superposition of simple vibrations. Fourier's theorem. — 

A particle may perform simultaneously two or more distinct 
vibratory movements. In such a case the vibrations are said to 
be superposed. Thus, if the hand be moved slowly up and 
down and if, at the same time, one finger be moved quickly up 
and down, the moving finger would trace a curve similar to the 
third or fourth or fifth curve in Fig. 431. In this example one 
vibration is assumed to be of low frequency and the other of high 
frequency in order that the movements may not be too greatly 
confused ; as a matter of fact, however, any number of vibratory 
movements, whatever their amplitudes and frequencies, may be 
performed by a particle simultaneously. 

Fourier's theorem. — Any periodic vibration of frequency n, 
however complicated, may be matched by superposing simple 
vibrations of which the frequencies are n, 2w, 3^, 4^, and so 
on, provided the respective amplitudes are properly chosen. 
That is, any given compound vibration of frequency n may 
be thought of as composed of a series of superposed simple 
vibrations of which the frequencies are n, 2n, 37^, 4^, and so on. 

Application of Fourier's theorem to the explanation of tone 
quality or timbre. — Aside from accompanying noises which fre- 
quently characterize musical tones, the difference in quality 
between two musical tones depends upon the character of the 
oscillations. If the oscillations are simply harmonic, the tone is 
called a simple tone. If the oscillations are compound, the tone 
is called a compound tone or clang. According to Fourier's 
theorem, a compound tone is the blending together of a series of 
simple tones of which the frequencies are n, 2n, 3^2, 4^, and so 
on. The tone of lowest frequency n is called the fundamental 
and the other tones are called overtones. The overtones in the 
note which is produced by a piano string can be heard easily and 
distinctly by a musically trained ear. 



TONES AND NOISES. 493 

355. Simple and compound wave-trains. Fourier's theorem. 
— ^When a wave-train passes through a medium, each particle of 
the medium oscillates. When each particle of the medium per- 
forms simple harmonic motion during the passage of a wave- train, 
the wave- train is called a simple wave-train. A simple wave-train 
is represented graphically by a curve of sines. 

When, during the passage of a wave-train, the particles of the 
medium perform periodic movements which are not simply har- 
monic, the wave -train is called a compound wave-train. A com- 
pound wave-train is represented graphically by a periodic curve 
which is not a curve of sines. Thus, the sine curve in Fig. 432 
may be thought of as representing a simple wave-train, and the 
periodic curve in Fig. 433 may be thought of as representing a 
compound wave- train. 

Fourier's theorem. — The periodic curve In Fig. 433 may be 
matched by superposing a series of sine curves of which the 
wave-lengths are X, X/2, X/3, X/4, etc., where X is the wave- 
length of the curve in Fig. 433.* Therefore, a compound wave- 
train may be thought of as made up of a series of simple wave- 
trains of which the wave-lengths are as above specified. It is for 
this reason that a wave-train which is not represented by a curve 
of sines is called a compound wave-train. 

A body which performs simple vibrations sends out a simple 
wave-train of sound waves, and a body which performs compound 
vibrations sends out a compound wave-train of sound waves. 

* This statement of Fourier's theorem is substantially identical to the statement 
given in Art. 354. 



ii 



CHAPTER XXIX. 

FREE VIBRATIONS OF ELASTIC BODIES. 

356. Vibration of a plucked string (special cases). — When 
a string is pulled to one side and released it vibrates in a manner 
indicated in Fig. 434. The shape of the string just before it is 




Fig. 434a. 

released is shown in Fig. 434a. When the string Is released a 
straight and uniformly downward moving portion develops more 




W ^-^"^" moving '^^^•~. ,^ 

TTTTTTTTTTTT 




Fig. 434b. 

and more as indicated In Fig. 434^ until the whole string Is 
moving downwards as shown In Fig. 434c. Then the moving 



-.- — " moving moving moving -^..^ 

ZfS-^i i M i I 4 I i 1 i i i 1 i 1 i i i i J^K-A 



Fig. 434c. 
494 



FREE VIBRATIONS OF ELASTIC BODIES. 



495 



portion grows shorter and shorter, as indicated in Fig. 434<i, 
until the whole string is standing still as shown in Fig. 434^. 




W moving W 



Fig. 434d. 




The string then moves upwards through the same series of stages 
in reverse order until it comes back to its initial condition as 
represented in Fig. 434a; and so on, over and over again. The 




Fig. 4346. 

sharp bends or corners WW travel to right or left along the 
string at velocity V which is given by the equation 



V 



-4 



T 

m 



(109) 



where T is the tension of the string in dynes (or poundals) and 
m is the mass in grams (or pounds) of one centimeter (or foot) 
of the string.* 

When a string Is plucked near one end it moves as indicated in 
Fig. 435. A uniformly moving portion develops after the string 
is released, this moving portion is straight (as in Fig. 434) and 
equally inclined to the two portions AB and CB, and as this 
straight portion travels sidewise it carries the string through a 

* This equation is discussed in Art. 359. 



496 



THE THEORY OF SOUND. 



series of configurations from the initial configuration ABC to 

ADC and back again. 

B 

A 




Fig. 435. 

If a string is plucked as indicated by the full line in Fig. 436, 
two straight moving portions develop after release, as indicated, 
and the middle point n of the string never moves. 

It is evident from the above discussion that the plucked string 
of a guitar or mandolin may vibrate in an endless variety of 




ways, depending upon the initial distortion produced by the 
operation of plucking. Also the bowed string of the violin and 
the hammered string of the piano can each vibrate in an endless 
variety of ways.* 

* A very simple elementary discussion of the vibration of strings is given in 
Franklin, MacNutt and Charles' Calculus, pages 190-209; published by the 
authors, South Bethlehem, Pa. 

See also a very simple discussion by W. S. Franklin in the Journal of the Franklin 
Institute, Vol. 179, page 559. 

A good discussion of this subject is given in Byerly's Fourier's Series and Spherical 
Harmonics, Ginn & Co., 

A mathematical discussion of the bowed string of the violin is given in an Ap- 
pendix of Helmholtz's Tonempfindungen (translated by Alexander J. Ellis as Sensa- 
tions of Tone, Longmans, Green & Co.). 

A beautiful experimental study of the motion of bowed strings is to be found 
in a paper by Krigar-Menzel and Raps, Wiedemann's Annalen, Vol. 44, page 623. 



FREE VIBRATIONS OF ELASTIC BODIES. 497 

357. Vibration of an air column (special case). — Consider a 
long tube closed at both ends; and imagine the air in one end 
of the tube to be slightly compressed, and the air in the other 
end of the tube to be slightly rarefied as indicated in the upper 
half of Fig. 437. When the gate valve is suddenly opened the 



rarefied air (still) " 


' compressed air (stilD 


|yX('>:V';.v':*-':^ :":•;:•;':•/ ;^ 





^ :•^^^•V::^^v:A■':•::0:v:v;i ' • ^ 

G 



rarefied air {still) W moving 



4 



air . W compressed air (still) 



B, 



■ <„ ■ normal pressure » ^ 

Fig. 437. 

air in the tube surges back and forth along the tube" in a manner 
which is mathematically identical to the motion of a string 
which is plucked at its center. A uniformly moving body of 
air at normal atmospheric pressure develops as indicated in the 
lower half of Fig. 437, and the successive stages of a complete 
oscillation may be understood by a proper interpretation of Figs. 
434 in which the shape of the string is to be thought of as a 
curve whose slope (positive or negative) represents compression 
or rarefaction of the air at each point of the tube, and whose 
upward or downward velocity v at any point represents the 
velocity of the air to right or left along the tube at that particular 
point. 

After the gate valve G in Fig. 437 Is opened, the air at the 
middle of the tube is always at normal atmospheric pressure, and 
the air in each half of the tube oscillates as if the tube were cut in 
two in the middle and each half left open to the air. Thus Fig. 
438 shows eight successive stages of one complete oscillation 
of the air in a tube in which air is initially compressed and one 
end of the tube opened suddenly. 

B2 shows uniform motion to the left and normal atmospheric 
pressure being established as the wave of starting W travels to 
the right at the velocity of sound ; 
33 



498 



THE THEORY OF SOUND. 
compressed air (siilf) 



^iiiiJi^iti^i^iiii^ii^iJiiiiiijiiiiiiiijiiiiui^^ 



B, 



moving air 



W 



compressed air 






^s^ 



normal pressure y ^ 

moving air 



still 



moving air 



normal pressure 



^3 



rarefied air 






»4 



normal pressure '^ y 

rarefied air 



still 



% 



moving air 



still 



rarefied air 






B. 




normal pressure 



moving air 



moving air 



normal pressure 

W'" compressed air 



B, 






Bf> 



normal pressure 



still 



compressed air 







Be 



still 

Fig. 438. 



FREE VIBRATIONS OF ELASTIC BODIES. 499 

Bs shows the state of affairs when the wave of starting W has 
reached the closed end of the tube; 

Bi shows a region of rarefied still air being established as the 
wave of arrest W travels to the left at the velocity of sound ; 

Bf, shows the state of affairs when the wave of arrest W has 
reached the open end of the tube; 

B^ shows uniform motion to the right and atmospheric pres- 
sure being established as the wave of starting W" travels to the 
right at the velocity of sound ; 

B^ shows the state of affairs when the wave of starting W 
has reached the closed end of the tube; 

Bs shows a region of compressed still air being established as 
the wave of arrest W" travels to the left at the velocity of sound ; 
and 

B^ shows the air in the tube brought back to its initial condi- 
tion. 

358. Simple modes of vibration. — In the above described oscil- 
lations of strings and air columns each particle of the string, or air, 
performs periodic motion which is very far from being simple har- 
monic motion. It is desirable, however, to consider those modes 
of oscillations of a string, or air column as fundamental and 
simple in which each particle of the string, or each particle of 
the air does perform simple harmonic motion. Such a mode of 
oscillation of a string or air column is called a simple mode. 

It is evident, from the discussion in Arts. 356 and 357 that 
the oscillation of a string or air column involves wave motion 
along the string or air column, and the discussion of what are 
called simple modes of oscillation may be carried out most easily 
by considering the passage of simple wave-trains along the string 
or air column, but before this can be done some preliminary 
discussion is necessary as given in the three following articles. 

359. Discussion of a wave on a stretched string or wire. — 
Figure 439 represents a belt encircling a pulley of which the radius 
is r. The tension T of the belt produces a certain resultant 
inward pull on each centimeter of the belt; and if the pulley is 



500 



THE THEORY OF SOUND. 



\U=4/ ; 



belt 



rotating, each particle of the belt will be traveling in a circular 
orbit and must necessarily be pulled inwards by an unbalanced 
force. It is reguired to find the velocity V at which the belt must 

travel around the pulley in order that the 
restdtant inward pull on each centi- 
meter of the belt due to the tension T 
of the belt may be just sufficient to pro- 
duce the inward acceleration of each 
particle of the belt as it circles round the 
pulley. 

When the pulley Is standing, still 
the outward push of the pulley on 
each centimeter of the belt must bal- 
ance the resultant inward pull on 
each centimeter of the belt due to the 
tension T of the belt. Indeed, the 
forces which act on the semi-circum- 
ference of the belt are exactly like the 
forces which act upon the half-ring which is shown in Figs. 58 
and 59 of Art. 52. Therefore from equations (25) of Art. 52 
we have: 




2r- 



Fig. 439. 



7^ = Tjr 



(i) 



where T is the tension of the belt in Fig. 439, r is the radius of 
the pulley, and F is the resultant force (due to .7") pulling 
inwards on each centimeter of the belt. 

Let V be the velocity of pulley rim and belt, then V^jr is 
the inward acceleration (towards the center of the pulley) of each 
particle of the moving belt according to equation (8) of Art. 31. 
Let m be the mass in grams of each centimeter of the belt then 
mV^/r is the unbalanced inward force which must act on each 
centimeter of the belt to produce the acceleration V^/r, and if 
the belt tension barely supplies this force (without pushing the 
belt against the pulley face) we must have 

T mV^ .... 



FREE VIBRATIONS OF ELASTIC BODIES. 



501 



This equation may be used to determine the velocity of a wave 
on a stretched string or wire as follows: Let AB, Fig. 440 be a 




Fig. 440. 

wire of which the mass of each centimeter is m grams, and which 
is under a tension of T dynes. Imagine the wire to be drawn at 
velocity V through an irregularly bent guide tube, and let it 
be assumed that the wire slides through the tube without 
friction. Then the wire would not he pulled against the side of the 
tube anywhere by its tension nor thrown against the side of the tube 
anywhere by centrifugal action if equation (ii) were satisfied, that 
is, if the velocity of the wire were 



7= a/- 

m 



(109) 



Therefore, the bend once formed on the moving wire would be 
permanent even if there were no guide tube for the wire to pass 
through. Such a standing bend on a moving wire is a wave 
traveling along the wire at velocity V as given by equation (109). 

360. Standing wave-trains. — When one end of a stretched rope 
is moved rapidly up and down, the tube quickly settles to a 
steady state of oscillation in which a series of points n along the 
rope remain stationary, while the intervening portions of the 
rope surge up and down, as indicated in Fig. 441. The heavy 
line in Fig. 441 shows the position of the rope at a given instant; 




moving up moving down moving up 

Fig. 441. 



moving down 



502 



THE THEORY OF SOUND. 



a snap-shot of the rope, as it were. This oscillatory motion, 
which is entirely devoid of progressive character, is called a 
standing wave-train. The stationary points n are called nodes, 
the intervening portions of the vibrating rope are called vibrating 
segments, and the middle point of a vibrating segment is called 
an antinode. It is important to keep in mind the distinction 
between an advancing wave-train (which is usually called, simply, 
a wave-train) and a standing wave-train. In an advancing wave- 
train no portion of the rope remains stationary, in fact, every 
particle of the rope moves in precisely the same way and to 



o 




|p 

k 


|q 




1^ 


[Urn 


[p 
I 


\ r 


;n 


A 
B 


1 ^^^k_^ 


^ 


1WW 




1 A 
B 


\ 




1 

1 

1 

1 
1 
1 
1 

1 
1 


u ! \ 


— 1 


1 

R 





N 


J/ • ' 


r 


I 1 


\ 

p 






p 


]g. 




\ 

P 


sf 



Fig. 442. 

exactly the same extent, but not simultaneously, each succeeding 
particle being a little later in its movements. In a standing 
wave -train, on the other hand, the rope does not move at all at 
the nodes, the amplitude of motion is a maximum at the anti- 



FREE VIBRATIONS OF ELASTIC BODIES. 



503 



nodes, and all the particles move simultaneously, that is, all the 
particles in a vibrating segment move up or down, together. 
The vibratory motion which is represented in Fig. 441 may be 
resolved into two oppositely moving (progressing) wave-trains. 
Consider two similar wave-trains A A and BB, Figs. 442, 443, 
444 and 445, moving in opposite directions as indicated by 




Fig. 443. 

the heavy arrows. These wave- trains are supposed to be travers- 
ing the same portion of the rope at the same time, and AA is 
drawn above BB merely to avoid confusion. The actual dis- 
placement of each particle of the rope is equal to the sum of 
the displacements of that particle due to each wave-train, and 
the actual velocity of each particle is equal to the sum of the 
velocities of that particle due to each wave- train. Now, the ordi- 
nates of the traveling curve AA, Fig.. 442, when they reach 
the point p as the wave-train A A moves to the right are at 



504 



THE THEORY OF SOUND 



each instant equal and opposite to the ordinates 0' of the traveling 
curve BB when they reach the point p as the wave-train BB 
moves to the left. Therefore the points ppp of the rope remain 
stationary. 

The regions between the points pp, on the other hand, move 
up and down (to right and left in case of an air wave in a tube) 
as the two wave- trains A A and BB travel through or over each 




Fig. 444. 

Other. The resultant of the two wave-trains AA and BB is 
therefore a standing wave-train with nodal points at ppp- 
Figure 442 shows the positions of the two wave-trains AA 
and BB and their resultant RR at a given instant. The small 
vertical arrows show the velocities of the various parts of the 
rope. Figure 443 shows the positions of the two oppositely 
moving wave-trains AA and BB and their resultant RR at 



FREE VIBRATIONS OF ELASTIC BODIES. 



505 



a later Instant when AA has moved one sixteenth of a wave- 
length to the right and BB has moved one sixteenth of a wave- 
length to the left. Figure 444 shows the position of A A and BB 
and their resultant RR (a straight line) at a still later Instant 
when AA has moved one eighth of a wave-length to the right 




• Fig. 445. 

and BB has moved one eighth of a wave-length to the left. 
The small vertical arrows show the velocities of the various parts 
of the rope as before. Figure 445 shows the positions of AA 
and BB and their resultant RR at a still later Instant when 
A A has moved three sixteenths of a wave-length to the right and 
BB has moved three sixteenths of a wave-length to the left. 

Note. — ^The distance from node to node (or from antlnode to 
antinode) In Figs. 442-445 Is equal to X/2, where X Is the 
wave-length of each of the advancing wave- trains A A and BB. 

Figure 446 shows the vibrating rope when It Is bent to the 



5o6 



THE THEORY OF SOUND. 



greatest extent 



At this instant the velocity (sidewise) of the 
rope is everywhere zero. Figure 447 shows the vibrating rope 




Fig. 446. 

when its sidewise motion is everywhere a maximum. At this 
instant the rope is straight as indicated by the Hne CD. 

(a) The rope 7iever moves at a node of a standing wave-train 
hut is, at times, abnormally stretched. The abnormal stretching at 
a node p is evident from Fig. 446 when we consider that the 



1 

1 

1 




J 




, 


9 


I 






\P 








1 
1 
1 

1 


Q 








1 

1 


1 


. 
















^ ! 1 1 


1 1 


















1 1 1 I 


CI 1 \ 


















\ l/> 






















* 






' 


' 






• 





Fig. 447- 

normal length of each portion of the stretched rope is the pro- 
jection of that portion on the straight line AB, so that the rope 
is most stretched where it is most inclined to this line. 

(Jb) The rope has a maximum of motion at an antinode q of a 
standing wave-train hut its amount of stretch is always normal. 
The normal amount of stretch of the rope at an antinode is shown 
by the fact that at an antinode it is always parallel to the line 
AB as shown in Fig. 441. 

For the case of a stationary or standing wave-train in the 
air in a tube the above propositions (a) and {h) become : 

(a) The air never moves at a node hut it is alternately compressed 
and rarefied. 



FREE VIBRATIONS OF ELASTIC BODIES. 507 

(b) The air at an ardmode has a maximum of motion hut it is 
always at normal pressure. 

361. Standing wave trains by reflection. — When a train of 
waves on a rope (or in an air tube) is reflected from one end of the 
rope (or air tube), the reflected train and the tail end of the 
original train travel simultaneously over the same portion of the 
rope (or air tube), and a stationary wave train is the result. 
We will discuss this phenomenon for the case of an air tube for 
the sake of generality. 

If the end of the tube where the reflection takes place is 
closed, then the air at the end of the tube cannot move but it 
can, of course, be compressed or rarefied. The reflection must 
take place so as to satisfy this necessary condition, and therefore 
a node of the standing wave- train is formed at the closed end of 
the tube. 

If the end of the tube where reflection takes place is open, 
then, of course, the air at the end of the tube can move but it 
cannot be compressed or rarefied. The reflection must take 
place so as to satisfy this necessary condition, and therefore an 
antinode of the standing wave-train is formed at the open end of 
the tube. 

362. Simple modes of vibration of strings. — Consider an in- 
definitely long stretched wire or string, AB, Fig. 448, fixed to a 

A B 



Fig. 448. 

rigid support at end B. Imagine a simple wave-train of wave- 
length X to approach the end B and be turned back or reflected. 
A standing wave- train will be formed as explained in Art. 361, 
and the nodes of this standing train will be at distances X/2 
apart, one node being at the rigid end B of the string. 

This standing wave-train once established, a rigid support 



508 THE THEORY OF SOUND. 

might be placed under the string at any one of the nodes, and the 
string between this new support and B would continue its vibra- 
tory motion unchanged, except, of course, that its motion would 
be slowly stopped by friction. Therefore, the length / of a 
vibrating string may be any multiple of X/2,* that is, 

/ = — * (1) 

in which n is any whole number. Let V be the velocity with 
which a wave-train travels along the stretched string, let r be 
the period of one oscillation of the string, and let / be the num- 
ber of oscillations per second (the frequency). Then, we have 

\= Vt (ii) 

as explained in Art. 248. Substituting this value of X in equa- 
tion (i) and solving for r, we have 

or since f = i/r, we have 

uV 
J = -I (no) 

This equation expresses the frequency of oscillation of a string 
(vibrating in a simple mode) in terms of the velocity of transmis- 
sion Ft of waves along the string, and the length / of the 
string, n being any whole number. 

When n is unity, the whole string is one vibrating segment, 
the string vibrates in what is called its fundamental mode, and 
gives what is called its fundamental tone. When n = 2, the 
string vibrates in two segments, and gives what is called its 
secondX overtone. When ^ = 3, the string vibrates in three 
segments and gives what is called its third overtone, etc. 

* This is merely a brief way of saying that the half -wave-length X/2 may be any 
aliquot part of Z, for the length of the string is given in any particular case. 

t See equation (109), Art. 359. 

X This is really the first overtone, considering that the fundamental is not, 
properly speaking, an overtone; but it is convenient to designate the order of the 
overtones by the corresponding values of w. *• 



FREE VIBRATIONS OF ELASTIC BODIES. 509 

A string may be made to vibrate approximately* in a funda- 
mental mode by drawing a violin bow across it near one end and 
touching it with the finger at a, as shown in Fig. 449. This 




>V}JMJPJJ^. 



Fig. 449. 

figure shows the string vibrating in three segments [w = 3 in 
equation (no)]. In this case a light paper rider remains on the 
string at the node n, whereas such a rider is thrown off the 
string at other points as indicated in the figure. 

When a string is plucked, or struck with a hammer, or stroked 
with a violin bow, it performs simultaneously its various simple 
modes of oscillation, and gives its fundamental tone together with 
its various overtones. The relative intensities of the funda- 
mental and various overtones depend upon the position of the 
point where the string is plucked, or stnick, or bowed. Thus a 
guitar or mandolin string gives a quality of tone which varies 
quite perceptibly with the location of the point where the string 
is plucked. In the piano, the best quality of tone is produced 
when each hammer strikes its string at a point about one seventh 
of the length of the string fro^n one end, except in case of the very 
short strings where the hammers must strike near the middle of 
the strings to give tones of the requisite loudness. 

363. Simple modes of vibration of air columns. — ^Waves on 
wires, such as are described in this text are called transverse 
waves because the parts of the wire move sidewise as the waves 
pass by. Waves in air and, in particular, waves in an air tube 
are called longitudinal waves because any portion of the air in 

* In Fig. 449 the string vibrates simultaneously in all those simple modes for 
which a node exists at the point a. In the particular case represented in Fig, 449 
the 3d, 6th, 9th, I2th, etc., modes are represented in the vibrations of the string. 



510 



THE THEORY OF SOUND. 



the tube moves back and forth along the tube (in the line of 
travel of the waves) as the waves pass by. 

Consider an indefinitely long tube AB, Fig. 450, closed at one 




Fig. 450. 

end B. Imagine a simple wave-train of longitudinal waves, of 
wave-length X, to approach the closed end B, in the tube. This 
wave- train will be reflected at B, the reflected train and ad- 
vancing train will together form a standing wave- train, and the 
nodes of this standing wave-train will be at a distance X/2 
from each other as shown in Fig. 450, the closed end of the tube 
being a node. 

Once this standing wave-train is established, an air-tight 
gate-valve might be placed in the tube at any node, and the air 
between the gate and the closed end B .would continue its vibra- 
tory motion unchanged, except, of course, its motion would die 
away because of friction. Therefore, (i) the length I of a vibrating 
air column which is closed at both ends may be any multiple of X/2.* 

The air pressure at any antinode in Fig. 450 is invariable 
and equal to atmospheric pressure, but the air at an antinode 
surges back and forth along the tube. Such motion is evidently 
possible at an open end of the tube, and at such an end the pres- 
sure would necessarily be the same as atmospheric pressure. 
Therefore, the standing wave- train being once established, (2) 
the tube might be cut off and left open at any antinode a and 
the air would continue to vibrate in the cut-off portion aB 
(open at a closed at B), or (3) the tube might be cut off and 

* This is a brief way to say that the half-wave-length X/2 may be any aliquot 
part of I, for the length of the air column is given in any particular case. 



FREE VIBRATIONS OF ELASTIC BODIES. 511 

left open at any two antinodes and the air would continue to 
vibrate in the cut-off portion ah (open at both ends). In case 
(2) the length of the vibrating air column is an odd number of 
quarter wave-lengths X/4; and in case (j) the length of the vibrating 
air column is an even number of quarter wave-lengths or any multiple 
of X/2. 

Therefore in cases (i) and (3) we have: 

7 ^^ / t. f closed 1 ^ ^ , \ /.x 

/ = — I tube i ^ at both ends 1 (l) 

2 \ I open j J ^ ^ 

in which n is any whole number. Substituting for X the value 
Vt, where V is the velocity of sound and r is the period of 
one oscillation of the air column, and solving for r we have 

2I / ( closed 1 .. .. ^ \ /..x 

r = 1 1 tube < (-at both ends 1 { U ) 

nV\ I open J / ^ ^ 

or, since / = i/r, we have 

X ^^ f u f closed] u u . \ / N 

T = ( tube < > at both ends ) (III) 

•^ 2/ \ I open j / ^ ^ 

This equation expresses the frequency of oscillation of an air 
column (closed at both ends or open at both ends) in terms of 
the velocity of sound V and the length / of the column, n 
being any whole number. 
In case (2) we have 

/ = (tube closed at one end) (ui) 

4 

in which n^ is an odd number. Substituting for X the value 
Vt, where V is the velocity of sound and r is the period of 
one oscillation of the air column, and solving for r, we have 

(tube closed at one end) (iv) 



n'V 



or, since f = - , we have 



n'V 

f = — — (tube closed at one end) (112) 

4/ 



512 



THE THEORY OF SOUND. 



This equation expresses the frequency of oscillation of an air 
column (closed at one end) in terms of the velocity of sound V 
and the length / of the column, n^ being any odd number. 

A very beautiful experiment showing the vibration of an air 
column in various simple modes is the following: An ordinary 
whistle mouthpiece is fixed to the end of a long glass tube. A 
weak blast of air causes the whistle to sound a low-pitch tone, 
and with increasing air pressure, the air column in the tube breaks 
up into a greater and greater number of vibrating segments, giv- 
ing a higher and higher tone. These vibrating segments may be 
rendered visible by placing lycopodium powder in the tube and 
maintaining a sufficiently constant air pressure to give an invari- 
able tone on the whistle. The air, as it surges back and forth 
(parallel to the length of the tube) in the vibrating segments, 
sweeps the lycopodium powder into small heaps at the nodes. 
It is best in this experiment to have the end of the tube closed 
to prevent violent air currents, and to use dry air in order that 
the lycopodium may not become moist and adhere to the walls 
of the tube. 

364. The organ pipe is a device, very similar to the bark 
whistles known to children, in which a column of air is set into 
vibration and caused to give a musical tone. Sectional views of 




Fig. 451- 

complete organ pipes are shown in Figs. 452 and 453, but Fig. 
451 shows the details of construction and action. Air under 
moderate pressure enters at the tube /, flows through the chamber 
c, issues as an air blast b, and strikes against the sharp lip /. 
The quavering of this air blast starts the air column vibrating 
feebly, these vibrations react upon the air blast and cause it to 
play from one side to the other of the lip /, and this effect re- 



FREE VIBRATIONS OF ELASTIC BODIES. 



513 



inforces the vibrations so that they quickly become energetic, 
and they are sustained as long as the air blast continues. 




open air column ooen 



fundamental mode /isJ. in equation (vii) 




vy/////y/MM/^///M//ff/////i/fj//j///////////7mM//////a//f//M//UM/////um/M^^ 



second mode n^ in equation {vii) 




^.,rffZ;7777Z7 y/MMM//mJW WM/JJWJ///W////m^^^^ 



'»y'yy^yryyj!fr^^^X»7XV>'yy/!aM'^^^ 



third mode n=5 in equation (vii) 

Fig. 452. 

The dotted curves in Fig. 452 are intended to show the char- 
acter of the motion of the air in an organ pipe which is open 
at both ends, and the dotted curves in Fig. 453 are intended 
to show the character of the motion of the air in an organ pipe 
which is closed at one end. 

Figures 452 and 453 represent simple modes 01 oscillation of 
organ pipes. As a matter of fact, however, an organ pipe always 
performs its various simple modes of oscillation simultaneously, 
and the musical tone produced by an organ pipe is therefore com- 
posed of a series of simple tones blended together. With broad 
short pipes the overtones are usually very weak, and the sound 
approaches what is called a simple tone in character. With long 
narrow pipes, the overtones are more pronounced; and when a 
long narrow organ pipe is blown strongly a single one of its 
overtones is usually produced, the others remaining inaudible. 

The clarinet, the flute, the cornet and the bugle are examples 
of musical instruments in which tones are produced by the vibra- 
34 



514 



THE THEORY OF SOUND. 



tion of air columns. In the clarinet and the flute, the length 
(and thereby the pitch) of the vibrating air column is altered by 



open air column closed* 




fundamental mode n'= 1 in equation (x) 




.r-rr / T /r^7///TJ/.')>7A^ - ^/7/////y/>///y>//////^y'/////y/?>/^/^//.'^^^^^^^^J^^. 



- -—--'I 



mode corresponding to n'=z3 in equation (x) 




..^MM?/>^j/^^M^}MM^M;;;M^^/w;;M;wwmw;;/?;//?//,w;M//^///;M;?M, 



mode corresponding to n'=5 in equation (x) 

Fig. 453- 

uncovering openings in the side of the tube. In the cornet, the 
length of the vibrating air column is altered by including or 
excluding extra lengths of tube by means of valve connections. 
In the bugle, the length of the vibrating air column is fixed, and 
the various notes are obtained by causing the air column to 
vibrate in one or another simple mode; therefore the only tones 
which can be produced on a bugle are the tones whose vibration 
frequencies are proportional to the numbers i, 2, 3, 4, 5, 6, 7, 8, 
etc. In fact, the tones 3, 4, 5 and 6 are the ones ordinarily 
employed in the bugle. 

365. Vibrations of rods. — The longitudinal vibrations of rods 
are exactly similar to the longitudinal vibrations of air columns 
and need not be discussed. When a rod is struck on one side 
with a hammer, it is caused to vibrate with a sidewise or trans- 
verse motion. One can always hear a number of distinct tones 
in the sound produced by a vibrating rod when the vibrations 
have been produced by a hammer blow, and each of these tones 



FREE VIBRATIONS OF ELASTIC BODIES. 515 

is due to a simple mode of vibration of the rod; a simple mode 
of oscillation being a type of oscillation in which every particle 
of the rod performs simple harmonic motion of a certain fre- 
quency. 

In the case of vibrating strings and air columns, the vibration 
frequencies which correspond to the various simple modes of 
oscillation are approximately in proportion to the successive 
whole numbers, that is to say, the tones constitute what is called 
a harmonic series. In the case of rods, however, the vibration 
frequencies which correspond to the various simple modes of 
transverse oscillation do not form a simple series of numbers. 
Thus, Fig. 454 represents three successive simple modes of oscilla- 
tion of a long steel rod.* The frequencies of the successive 
simple modes are given in terms of the frequency of the funda- 
mental mode which is taken arbitrarily as i,ooo. The funda- 
mental mode has two nodes, the second mode has three nodes, 

f-c^.. .,-n 



1 


-^ ^- 




•" 


"Z. —" ' 


" *" 




'~ •- ^ "^ 


- 




^ ^^ " ^ 




<<- — 


-"" 


j 
— ^ — 


**■ 


'^ ^^ 


-^ - 





"_'-'--■-"' 


'-' 


-^•^i C '■ 


^» 


1 




'6.224i 










0.5521 






■-■K 


0.2241 


— > 



first or fundamental mode; frequency taken as 1000 






^oJm^ asm "^ 0:3681 ^"6:i32i'^ 



c 

6,1321^ 0. 

second modei frequency 2754 



^~<r > 



''b.om^ 0.2621 ^ ~6:288l ^ 6:2621 ^o.om 

third mode j frequency 5402 

Fig. 454- 

* This figure is based on a theoretical paper by Seebeck which was published in 
1848. See Winkelmann's Handbuch der Phystk. 



516 



THE THEORY OF SOUND. 



3 C 




\ 

\ 
1 


1 \ 
1 1 






1 
1 


1 I 

1 


ly 


p 


n 


W 



/ 



A 



and the third mode has four nodes. The locations of the nodes 
are indicated in terms of the length / of the rod. In order to 
cause a rod to vibrate according to one of the sketches in Fig. 

454, the rod v/ould have to be supported at one or the other set 
of nodes and excited by a violin bow. The positions of the nodes 
and the relative frequencies of the various modes depend to some 
extent upon the ratio of length to diameter of rod. 

The tuning fork is a stiff rod bent into the form shown in Fig. 

455. The character of its fundamental mode of vibration is 

shown by the dotted lines. The two 
nodes nn are near together, and the 
intervening segment, together with the 
metal post P, move up and down 
through a small amplitude and cause 
the sounding board upon which the fork 
is mounted to vibrate in unison with 
the fork. The second simple mode of 
vibration of a fork usually gives a tone 
two or three octaves above the funda- 
mental, and it dies out quickly after the 
fork is set into vibration by a hammer 
blow, leaving the fundamental alone. The tuning fork, there- 
fore, gives a simple tone. 

366. Vibrations of plates. — ^When a plate, such as a circular 
saw of steel, is struck on one side with a hammer, it is set into 
vibration, and one can always hear a number of distinct tones 
in the sound produced. Each of these tones corresponds to a 
simple mode of vibration of the plate, a simple mode of oscillation 
being a type of oscillation in which every particle of the plate 
performs simple harmonic motion of a certain frequency. The 
vibration frequencies which correspond to the various simple 
modes of oscillation of a plate do not form a simple series of 
whole numbers. 

An elastic plate may be made to perform a simple mode of 
oscillation by supporting it in a clamp as shown in Fig. 456, 



w 



Fig. 455. 



FREE VIBRATIONS OF ELASTIC BODIES. 



517 




Fig. 456. 







Fig. 457- 



5i8 



THE THEORY OF SOUND. 



touching the fingers lightly against the plate in the positions 

(found by trial) corresponding to the nodes, and drawing a 

violin bow across the edge of the plate. When sand is strewn 

upon the plate, it is thrown away from the vibrating segments 

and heaped along the nodal lines. Thus Fig. 457 shows four 

sand figures obtained in this way. These figures were first 

studied by the Italian physicist Chladni and described in his 

treatise on acoustics in 1787. 

The bell may be looked upon as a cup-shaped plate. When a 

bell is struck, a number of distinct tones may be heard. These 

tones correspond to simple modes of vibration of the bell, and 

their frequencies are not usu- 

VeryWeak (480) 

Strong Tremulo(570 

Weak (372) 




Very Strong (240) 



^ 



ally in proportion to the whole 
numbers i, 2, 3, 4, 5, etc. 
Bell founders have, however, 
learned to produce bells of 
such shape as to make the 
more prominent tones har- 
monic; such a bell gives a 
rich and pleasing tone. Fig- 
ure 458 shows the more promi- 
nent tones of a Russian bell in the library of Cornell University. 
The numbers represent the number of vibrations per second. 
The tone marked "tremulo" consists of two tones differing very 
slightly in pitch, but of nearly the same loudness, and the tremu- 
lous character is produced by the beats of the tones. 



-Very Weak (186) 
-Very Weak (120) 



Fig. 458. 



CHAPTER XXX. 

FORCED VIBRATIONS AND RESONANCE. 

367. Free vibrations and forced vibrations. — The vibrations 
which a body performs when it is disturbed in any way and left 
to itself are called its free or proper vibrations. When a simple 
wave-train of sound waves of any wave-length strikes a body, the 
body is made to vibrate in unison, or in the same rhythm, with 
the impinging waves.* Such vibrations are called forced or im- 
pressed vibrations. A compound wave-train causes a body to 
perform simultaneously the various simple vibrations which cor- 
respond to the various simple wave -trains which enter into the 
composition of the compound wave- train. 

368. Damping. — Vibrations are said to be damped when they 
die out quickly. This damping effect is due in part to the dissi- 
pation of energy in the body in the form of heat, as it is repeat- 
edly distorted, and in part to the giving up of energy to the sur- 
rounding air. Thus, the vibrations of a light body which ex- 
poses considerable surface to the air, a diaphragm, for example, 
die out quickly: whereas, the vibrations of a heavy and highly 
elastic body, like a tuning fork, are but slightly damped. A 
heavy tuning fork performs several thousand perceptible vibra- 
tions when struck. The column of air in an organ pipe performs 
several hundred perceptible variations after the exciting cause 
ceases. A drum-head performs very few perceptible vibrations 
when struck. 

369. Resonance. — ^Very perceptible vibrations are impressed 

* This statement refers to the ultimate character of the motion and not to the 
motion which takes place during the time that the steady state of vibration is being 
established. Duiing the time that the steady state of vibration is being established, 
the body behaves as if its free vibrations and the ultimate forced vibrations existed 
simultaneously, and, as the free vibrations die out, the forced vibrations are left 
alone. 

519 



520 THE THEORY OF SOUND. 

upon a light diaphragm by sound wave-trains of any wave-length 
(frequency), but the vibrations which are impressed upon a heavy 
body, of which the damping is slight, are much more violent when 
the impressed frequency approaches the frequency of the free vibra- 
tions of the body. Thus, the sound of a tuning fork (removed 
from its sounding board) is perceptibly reinforced when it is held 
near the open end of a tube of any length; but if the length of 
the air column is adjusted, by pouring water into the tube for 
example, the sound becomes louder as the proper frequency of 
the air column approaches that of the fork; and it reaches a very 
distinct maximum of loudness when the impressed vibrations 
become proper to the air column. 

When the vibrations of a body are subject to very slight damp- 
ing, the impressed vibrations at proper frequency are very ener- 
getic as compared with the impressed vibrations of improper 
frequency. Thus, a massive tuning fork, mounted upon its 
sounding board, is thrown into very energetic vibration by a tone 
of its proper frequency sustained for four or five seconds, but it is 
scarcely affected at all by a tone of which the frequency differs 
very slightly from the proper frequency of the fork. This pro- 
nounced maximum violence of impressed vibration at proper fre- 
quency is called resonance, and the vibrating body or air column is 
called a resonator. 

When impressed vibrations are proper to a body, the impulse 
of each successive wave adds to the existing motion, and the 
vibrations increase in violence until the energy given to the 
vibrating body by the impinging waves is all dissipated by damp- 
ing. It is for this reason that the impressed vibrations become 
quite energetic when the damping is small. When, however, 
improper vibrations are impressed upon a body, the periodic 
forces with which the waves act upon the body must take the 
place, more or less, of the internal elastic forces which ordinarily 
cause the body to vibrate; consequently the vibrations cannot 
become very energetic. 

370. Analysis of compound tones by means of resonators. — 



FORCED VIBRATIONS AND RESONANCE. 521 

A resonator of which the proper frequency coincides with the fre- 
quency of one of the component tones of a compound tone or 
clang, responds to the component tone by resonance and in- 
creases its loudness. Thus, if the frequencies of the fundamental 
and successive overtones of a string are loo, 200, 300, 400, etc., 
vibrations per second, a resonator of which the proper frequency 
is 400 vibrations per second will be set into energetic vibration 
by the sound of the string, and this particular overtone will be 
increased in loudness. 

Any component tone of a clang may be easily detected, or 
brought to notice, by intermittently strengthening it by means 
of a resonator tuned to unison with it. A convenient resonator 
for this purpose is shown in Fig. 459. It consists of a hollow 
glass ball with an open mouth at a and an open tip h. The 
open tip h is inserted into the ear, and the sound waves act upon 
the inclosed air through the opening a. In order to analyze a 
clang, one after another of a series of such resonators must be 
applied to the ear. Figure 460 shows an adjustable resonator of 




ENCLOSED AIR 0PENIN6 



Fig. 459- Fig. 460. 

Koenig. It consists of two brass cups, one sliding into the other, 
and the outer cup ends in a small nipple which is inserted into 
the observer's ear. 

371. Vowel sounds. — Every vowel sound is characterized by 
one or two tones of definite pitch. The characteristic tones of 
some of the vowels, as determined by Helmholtz,* are shown 

* See Sensations of Tone (a translation of Helmholtz's Tonempfindungen by A. J. 
Ellis), pages 1 53-173. 

A very complete study of vowel sounds has been made by D. C. Miller. See 
pages 215-251, The Science of Musical Sounds, by D. C. Miller, The Macmillan Co., 
New York, 1916. 



522 



THE THEORY OF SOUND. 



in the following table, and Fig. 461 gives these characteristic 
tones in terms of the ordinary musical notation. 



K 



t t t 



t^t=; 



t 



r 



Fig. 461. 



Characteristic Tones of Vowel Sounds. 



Vowel. 


Tone. 


Vibration frequency.* 


u as in rude 


/ 


173 


6 as in no. 


c" 


517 


a as in paw. 


g" 


775 


a as in part. 


d^" 


1,096 


a as in pay. 


/and&&'" 


346 and 1,843 


e as in pet 


c'"" 


2,068 


e as in see. 


/ and d''" 


173 and 2,322 



In producing a given vowel sound, the mouth cavity is shaped 
so that the contained air has a proper frequency of vibration 
which corresponds to the characteristic tone of the vowel to be 
produced. The sound from the vocal chords is very complex (it 
contains tones of almost every pitch), and that particular tone 
which is in unison with the free or proper vibrations of the air in 
the mouth cavity, is greatly strengthened by resonance, thus pro- 
ducing the desired vowel sound. 

In ordinary speech, the sound produced by the vocal chords is 
very harsh and suffices to excite the resonance of the mouth 
cavity for the production of any vowel sound. The smooth tone 
of a singer, however, may not contain the characteristic tone of 
the vowel, so that this tone cannot be strengthened by resonance. 
In this case, those overtones of the note which is sung, which are 
nearest the characteristic tone of the vowel for which the mouth 

* Complete vibrations per second. 



FORCED VIBRATIONS AND RESONANCE. 523 

cavity is set, are slightly strengthened, and in this way the vowel 
sound is produced, although rather incompletely. It is a well- 
known fact that spoken words are much more easily understood 
than words which are sung, and the difference in distinctness is 
largely due to the imperfect character of the vowels when sung. 

Overtones near a given pitch are more widely separated in a 
note of high pitch than in a note of low pitch, so that the mouth 
cavity, shaped to give the characteristic tone of a vowel, is less 
likely to produce the desired effect with high notes than with low 
notes. Thus, the words of a soprano singer are less distinct than 
the words of a bass singer of the same schooling. 

The proper tones of the mouth cavity when it is shaped for the 
production of the vow^els o, a and a may be easily heard by 
thumping against the cheek when the mouth is prepared to sound 
these vowels. A very striking experiment is the following: The 
lungs are filled with purified hydrogen, and a series of words like 
rude, no, paw, part, pay, pet, see, are spoken very deliberately 
and carefully. The mouth cavity is automatically shaped to pro- 
duce the various characterizing tones, but the presence of hydro- 
gen instead of air in the mouth raises the pitch of the mouth 
cavity nearly two octaves and the result is that the attempt to 
produce the vow^els u, o, a, a, a, etc., fails in a most laugh- 
able manner. 

372. The phonograph. — A thin diaphragm carries a light tool 
which scratches a minute groove in a smooth rotating cylinder 
made of a hard wax-like compound. A sound striking the dia- 
phragm impresses vibrations upon it and causes the tool to cut a 
groove of varying depth. A record of the sound is thus made 
upon the cylinder. To reproduce the sound, a round-ended tool 
which is attached to the diaphragm is adjusted to follow the 
groove, and the varying depth of the groove lifts the tool up and 
down, causing the diaphragm to vibrate as before and the sound 
is reproduced. This description applies to the old cylinder type 
of phonograph, but the action of the disk type of phonograph is 
not essentially different. 



CHAPTER XXXI. 

THE EAR AND HEARING. 

373. The human ear. — It was pointed out by Helmholtz that 
our perception of the various simple tones in a clang must depend 
upon the existence of a series of organs (the end organs of the 
auditory nerves) in the ear, each of which has a proper vibration 
frequency and is sensitive (by resonance) to simple tones nearly 
in unison with it. Thus, a compound tone excites those particu- 
lar elements in the ear whose vibration frequencies correspond to 
the various simple tones of the clang. This action may be illus- 
trated by means of the piano as follows: A musical tone of 
characteristic quality, a vowel sound, for example, is sung loudly 
against the sounding board of a piano of which the dampers are 
raised so as to leave the strings free. Those strings which are 
capable of vibrating in unison with the various simple tones of 
the clang are set into vibration by resonance, and one can hear a 
continuation by the piano of the vowel sound after the singing 
ceases. Imagine each string of a piano to be connected to a 
nerve fiber, and we have an apparatus which would perceive 
tones as they are actually perceived by the ear. 

The elements in the ear which respond to tones by resonance 
are suppos-ed to be the shreds of the basilar membrane which are 
stretched across a long slender cavity called the cochlea. This 
cavity is coiled upon itself hke a small shell,* hence its name. 

374. Persistence of sound sensations. — When the stimulation 
of a nerve ceases, the accompanying sensation continues for a 
length of time which depends upon the intensity of the stimula- 
tion and which varies greatly with the different nerves. Sensa- 
tions of light persist much longer than sensations of sound. 

* See Sensations of Tone (Ellis' translation of Helmholtz's Tonempfindungen), 
pages 188-226. 

524 



THE EAR AND HEARING. 525 

Thus, an intermittent light gives a sensibly continuous sensation 
when the flashes follow one another at the rate of thirty or 
forty per second, whereas an intermittent sound, for example, 
the sound of a tuning fork of high pitch which is shut off from the 
ear periodically has been found by Mayer to give a continuous 
sensation when the frequency of intermittence reaches 135 per 
second. 

The persistence of sensations of light are shown by the fact 
that a rapidly moving spark looks like a streak of light, and the 
so-called strohoscopic effect depends upon the persistence of vision. 
Thus if a rotating wheel is illuminated by a series of quick' flashes 
of light the wheel either appears to stand still, or rotate slowly 
forwards, or rotate slowly backwards according as the time in- 
terval between the flashes is equal to, or slightly greater, or 
slightly less than the time for one spoke of the wheel to move 
forwards to the position of the next spoke. 

375. Consonance and dissonance.* — An intermittent or fluctu- 
ating tone produces an unpleasant sensation which is called 
discord or dissonance. A steady tone, or one which fluctuates so 
rapidly as to give a steady sensation, produces a pleasing effect 
which is called concord or consonance. These terms discord (or 
dissonance) and concord (or consonance) are used ordinarily to 
describe the effects produced by two or more simultaneous tones, 
that is, they are used to describe the relations of tones in music. 
Nevertheless, the above definitions are physically correct. 

Consider a tone which is intermittently shut off from the ear, 
the intermittence beginning at low frequency and increasing to 
greater and greater frequency. The dissonance at first increases 
with increasing frequency of intermittence, reaches a maximum, 
then falls off and disappears entirely when the frequency of inter- 
mittence becomes so great as to give a continuous sensation of 
tone. The frequency of intermittence for which the dissonance 

* The theory of consonance and dissonance as here developed is used by Helm- 
holtz in his very interesting theory of music. This theory is given in simple form 
in Chapter XVI of Franklin and MacNutt's Light and Sound, The Macmillian 
Co., 1909. 



526 



THE THEORY OF SOUND. 



is a maximum, and the frequency of intermittence for which the 
sensation becomes continuous, depend upon the vibration fre- 
quency of the tone which is used, as shown in the accompany- 
ing table, which is from experiments made by Mayer. 



Vibration frequency of 


Frequency of intermittence. 


tntermittent tone. 


When tone becomes smooth. 


When discord is a maximum. 


64 
128 
256 

384 
512 
640 
768 
1,024 


16 
26 

47 
60 

78 

90 

109 

135 


6.4 
10.4 
18.8 
24.0 
31.2 
36.0 
43-6 
54-0 



The fluctuations which produce dissonance in music are the 
fluctuations due to beats, which are described in the following 
paragraph. When two tones are in unison they give a smooth 
sensation. If the vibration frequency of one tone is slowly in- 
creased, beats occur with greater and greater frequency, the 
resulting intermittent sound produces a more and more discordant 
sensation which soon reaches a maximum, after which the discord 
decreases and finally disappears when the beats reach a suffi- 
ciently high frequency. 

376. Beats and combination tones. — Consider two simple 
tones of which the vibration frequencies , / and f, are nearly 
the same. At a certain instant the wave-trains which consti- 
tute these two tones will be in like phase as they enter the ear, 
giving a maximum loudness of sensation. When the tone of 
higher pitch has gained half a vibration (or half a wave-length) 
over the other, the wave-trains will be opposite in phase as they 
enter the ear, giving a minimum loudness of sensation. When 
the higher tone has gained a whole vibration over the other the 
waves will again enter the ear in like phase, and the loudness 
will again be a maximum, and so on. These periodic changes 
of loudness of two tones of approximately the same frequency 



THE EAR AND HEARING. 527 

constitute what are called heats. The number of beats occurring 
in one second is equal to / — f. 

Very prominent beats may be produced by sounding two sim- 
ilar organ pipes simultaneously, the open end of one of the pipes 
being partly covered to make it give a tone of slightly lower pitch 
than the other pipe. 

Combination tones. — The principle of superposition, namely, 
that two trains of waves may pass through the same region 
at the same time without affecting each other, or that an elastic 
system may perform a series of simple harmonic movements 
simultaneously without the various vibratory movements being 
affected by each other is no-t strictly true. The failure of this 
principle of superposition is due to what may be termed the 
incomplete elasticity of ordinary substances, that is, to the fact 
that the elastic forces brought Into play by the sum of the two 
distortions is not exactly equal to the sum of the elastic forces 
brought into play by the two distortions separately. The prin- 
ciple of superposition fails most decidedly when an imperfectly 
elastic substance Is vibrating very violently. 

Two weak tones do not sensibly affect each other when they 
are transmitted simultaneously through the same region of air or 
when they act simultaneously upon any elastic or approximately 
elastic system, such as the transmitting mechanism of the ear. 
As the tones grow louder, however, certain other tones produced 
by their mutual action begin to be heard. These accompanying 
tones are called secondary tones or combination tones. The Imper- 
fectly elastic chain of bones In the ear has most to do with the 
formation of combination tones. Everyone, perhaps, has noticed 
the harsh rattle in the ear which accompanies the loud sound of 
a dinner bell. This Is due to the Incomplete elasticity of the 
tissues which connect the tiny bones together. Indeed, it Is a sort 
of rattling effect of the bones on each other, and it Is the same 
thing essentially as a combination tone, only greatly exaggerated. 

Difference tones. — The most prominent combination tone Is 
that of which the frequency is equal to the difference of the fre- 



528 



THE THEORY OF SOUND 



quencies of the two primary tones. This is called the difference 
tone of the first order. This difference tone forms difference tones 
with each of the primary tones and these are called difference 
tones of the second order, and so on. 

Summation tones. — Very much less prominent than the dif- 
ference tones is the combination tone of which the frequency is 
equal to the sum of the frequencies of the two primary tones. 
This is called the summation tone of the first order. Summation 
tones of the first order form summation tones with each primary 
tone and these are called summation tones of the second order, and 
so on. 

Example. — The easiest method, perhaps, to produce an audi- 
/ ble combination tone is to use two large steel 
ifc"' bars of which the vibration frequencies are, 
say, 500 and 600 vibrations per second, re- 
spectively. When these bars are properly 
suspended (at the nodes of the fundamental 
mode, see Fig. 454), and struck in quick 
succession with a heavy hammer, a very dis- 
tinct tone is heard of which the pitch is 100 vibrations per second. 
A single steel bar of which the section is of the shape shown 
in Fig. 462, gives a tone of low pitch when struck on one of the 
flat faces, and a tone of high pitch when struck in a direction at 
right angles to this. A hammer blow in the direction of the 
dotted arrow causes the bar to give both tones simultaneously, 
and the first-order difference tone may be heard distinctly 
throughout a large room. 



Fig. 462. 



APPENDIX A. 

PROBLEMS. CHAPTER I. 

1. An iron casting contains 1,540 cubic inches of iron, and the 
density of iron is 0.28 pound per cubic inch. What is the mass 
of the casting in pounds? 

2. The density of water is approximately 62.5 pounds per 
cubic foot, and a vessel holds 200 pounds of water. What is 
the capacity of the vessel in cubic feet? 

3. The vessel specified in problem 2 holds 165 pounds of oil. 
What is the specific gravity of the oil ; what is the density of the 
oil in pounds per cubic foot? 

Note. The specific gravity of a substance at a given temperature is the ratio 
of the masses of equal volumes of the substance and water at the given temperature. 

4. A body travels 200 feet straight-away in 5 seconds. What 
is the average velocity of the body during the five seconds? 

5. A body travels at a constant velocity of 250 centimeters 
per second along a straight path. How far will it travel in 10 
minutes? 

6. A train starting from rest reaches a velocity of 50 miles 
per hour (73 J feet per second) after 150 seconds. At what 
average rate does the train gain velocity during the 150 seconds? 
Ans. 0.4887 feet per vsecond per second. 

7. A train moving at a speed of 50 miles per hour (73J feet 
per second) is brought to rest by the brakes in 16 seconds. At 
what average rate does the train lose velocity during the 16 
seconds? 

8. The train specified in problem 6 has a mass of 700,000 
pounds. What is the average pull in poundals exerted on the 
train by the locomotive while starting, friction being neglected? 
Reduce the result to "pounds," assuming one "pound" of force 
to be equal to 32 poundals. Ans. 10,690 "pounds." 

35 529 



530 APPENDIX A. 

9. The train specified in problem 7 has a mass of 700,000 
pounds. What is the average backward force exerted on the 
train due to braking? Reduce the result to "pounds" as ex- 
plained in problem 8. 

10. The maximum tractive effort of a locomotive is, say, one 
tenth of the weight of the locomotive. The mass of the loco- 
motive is 300,000 pounds and the mass of the train 1,400,000 
pounds. Find the maximum acceleration of the train which the 
locomotive can produce on a level track, allowing 10 "pounds" 
of backward friction drag for each ton (2,000 pounds) of train 
and locomotive. Ans. 0.405 feet per second per second. 

Note. Of course the tractive effort of the locomotive must accelerate the loco- 
motive as well as the train. 

11. An elevator car has a mass of 2,500 pounds. The car 
gains full velocity of 8 feet per second upwards in 2J seconds 
after starting, (a) Calculate the average upward pull of the 
elevator rope on the car during the period of starting, {h) Cal- 
culate the upward pull of the elevator rope on the car after the 
car has reached full speed (constant speed). Ignore friction in 
both cases, and assume one "pound" of force to be equal to 
32 poundals. Ans. (a) 2,750 "pounds"; {h) 2,500 "pounds." 

Note. While the car is starting upwards the upward pull of the rope on the 
car exceeds the downward pull of gravity on the car by ma poundals, where m 
is the mass of the car in pounds and a is the upward acceleration of the car in 
feet per second per second. 

12. Calculate the upward pull of the rope on the car in the 
previous problem while the car has a downward acceleration of 
3 feet per second per second. Ans. 2,265.6 "pounds." 

13. A 160-pound man stands on an elevator platform. With 
what force does the platform push upwards on the man's legs 
{a) while the car is standing still or moving at constant speed, 
{h) while the car is being accelerated upwards at the rate of 3.2 
feet per second per second, (c) while the car is being accelerated 
downwards at the rate of 4.8 feet per second per second ? Express 
the results in "pounds." Ans. {a) 160 "pounds," (&) 176 
"pounds," {c) 136 "pounds." 



PROBLEMS. 531 

14. A spring scale registers a pull of 10 "pounds" when 10 
pounds of beef is hung from it. What pull would the scale 
register if the butcher were standing on an elevator platform with 
an upward acceleration of 8 feet per second per second? Ans. 
I2| "pounds." 

15. A cord is strung over a pulley. At one end of the cord 
is a ten-pound body and at the other end of the cord is an eleven- 
pound body. Neglecting weight and mass of cord and neglecting 
friction and mass of pulley, find the acceleration of both bodies 
and the tension of the cord, taking the acceleration of gravity as 
32 feet per second per second. Ans. 1.523 feet per second per 
second, tension of cord 10.476 "pounds." 

Note. Let T be the tension of the cord in poundals and let a be the accelera- 
tion of both weights. Then the unbalanced force acting on the ii -pound body is 
(11 X 32) — T and this force produces the acceleration a of this body. That is, 
we have: 352 — T = iia. Similarly the unbalanced upward force acting on the 
lo-pound body is T — 320 poundals and the upward acceleration of this body is a 
so that we have: T — 320 = loa. 

16. The acceleration of gravity at London is 32.16 feet per 
second per second and the "pound" of force is the pull of gravity 
on a one-pound body in London. Find the exact value in 
"pounds" of the gravity pull on a 550- pound body at a place 
where the acceleration of gravity is 31.96 feet per second per 
second. Ans. 546.2 " pounds." 

17. The armature of an electric motor runs at a speed of 900 
revolutions per minute, and when the switch is opened the motor 
stops in 15 vseconds. Find the average spin-deceleration of the 
armature while stopping. Ans. 6.28 radians per second per 
second. 

Note. Spin-deceleration means a negative spin-acceleration, or the rate of 
loss of spin-velocity. 

18. A string is wrapped around the axle of a wheel as shown 
in Fig. 34, and a pull of 10 "pounds" (320 poundals) exerted 
steadily on the string for 2 seconds gives to the wheel a spin- 
velocity of 75 revolutions per second. The radius of the axle 
(distance from axis to center-line of string) is 0.25 inch, (a) 



532 APPENDIX A. 

What is the spin-acceleration of the wheel during the two seconds? 
(b) What is the value of the torque which is producing the spin- 
acceleration? (<:) What is the value in f.p.s. units of the spin- 
inertia of the wheel? Ans. (a) 235.6 radians per second per 
second; (b) 6.67 poundal-f eet ; (c) 0.0283 f.p.s. units [pound- 
(feet)2]. 

Note. The torque which acts on the wheel and axle is equal to Fr, where F 
is the pull of the string and r is the distance from the axis of the wheel to the 
center line of the string. 

19. A bicycle wheel has a spin-inertia of, say, 5 f.p.s. units 
[5 pound- (feet)^] and it is set rotating at a spin-velocity of 3 
revolutions per second and left to itself. The friction at the 
bearing brings the wheel to rest in 20 seconds, (a) What is the 
average rate at which the wheel loses spin-velocity while stop- 
ping? (b) What is the average value of the friction torque 
which stops the wheel? Ans. (a) 0.942 radian per second per 
second; (b) 4.71 poundal-f eet or 0.147 " pound "-feet. 

20. A falling ball passes a given point at a velocity of 12 feet 
per second. How far below the point is the ball after 5 seconds? 
How far does the ball fall during the fifth second after passing 
the given point? Air friction neglected. Ans. After 5 seconds 
the ball is 460 feet below the point, during the fifth second the 
ball falls 156 feet. 

Note. After 5 seconds the body will have gained 160 feet per second of velocity 
so that the initial velocity is 12 feet per second, the final velocity is 172 feet per 
second, the average velocity is t}(i2 + 172) feet per second, and the distance 
traveled during the 5 seconds is found by multiplying this average velocity by 
5 seconds. The argument of the second part of the problem is essentially the 
same as this. 

21. A heavy iron ball is tossed at a velocity of 20 feet per 
second in a direction 30° above the horizontal. What are its 
horizontal and vertical distances from the starting point after f 
second? Air friction neglected. Ans. Horizontal distance 12.99 
feet; vertical distance — 1.5 feet. 

Note. Find vertical and horizontal components of the initial velocity. The 
latter component remains unchanged while the vertical motion of the ball is pre- 
cisely what it would be if it had no horizontal motion. 



PROBLEMS. 533 

22. A lo-pound hammer, moving at a velocity of i6 feet per 
second, strikes a spike, and the spike is pushed one-half an inch 
into the beam into which it is being driven. What average force 
is exerted on the spike by the stoppage of the hammer? 

Note. Assume the hammer to lose velocity at a constant rate while stopping. 
Then the average velocity while stopping is |(i6 + o) feet per second, or 8 feet 
per second, and the time required for the hammer (and spike) to travel half an inch 
at an average velocity of 8 feet per second is 1/192 of a second. Therefore the 
hammer loses its velocity of 16 feet per second in 1/192 of a second which is at the 
average rate of 3,072 feet per second per second, and the force which is stopping 
the hammer (which is equal and opposite to the force exerted on the spike by the 
hammer) is equal to ^ X 10 X 3,072 "pounds." 

23. A 200-pound cannon ball having a velocity of 2,400 feet 
per second penetrates a clay bank to a depth of 12 feet in being 
stopped. Find the average force acting on the ball to stop it or 
the average force exerted on the clay bank by the ball. Ans. 
1,500,000 "pounds." 

24a. A train starts from station A and is accelerated at the 
rate of f of a mile per hour per second for 40 seconds. The 
train then runs at full speed of 30 miles per hour, and then it is 
decelerated at the rate of ij miles per hour per second until it 
stops at station B, The distance from station A to station B 
is 2 miles. Find (a) distance traveled while accelerating, {h) 
distance traveled while decelerating, (c) distance traveled at full 
speed, and {d) average velocity of whole run. Ans. (a) \ mile; 
(&) ^^ mile; {c) if miles; {d) 21.73 miles per hour. 

24b. An elevator cage has a mass of 2 ,500 pounds and its center 
of mass (which is also its center of gravity or point of application 
of the pull of gravity) is 2 feet to the right of the center of the 
cage. The cage is kept in a vertical position by guide rails at 
its sides, and we will assume that the car slides on these rails 
without friction so that each side rail pushes horizontally on 
the car. These two side pushes are exerted at upper right hand 
corner and lower left hand corner respectively, and the height 
of the car is 7 feet. Find the side push of each guide rail when 
the car has a downward acceleration of 8 feet per second per 
second. Ans. 535.7 "pounds." 



534 APPENDIX A. 

Note. Draw a sketch showing the outline of the car as a square and showing, 
as arrows, the four forces which act on the car. 

The car has translatory acceleration, only, and, according to Arts. 6 and 9, the 
unbalanced force acting on it is a downward force of 2500 X 8 poundals applied at 
its center of mass. Therefore if we subtract this unbalanced force from the total 
downward pull of gravity (2500 X 32 poundals) we have as a remainder a downward 
pull of 60,000 poundals, and the following forces which act on the car balance 
each other: (o) The downward force remainder of 60,000 poundals acting at the 
center of mass of the car, (b) The upward pull of the cable acting at the center of 
the car, (c) The force pushing to the left at the upper right hand corner, and (d) 
The force pushing to the right at the lower left hand corner. 

In the first place the forces a and h balance each other. Therefore the upward 
pull of the cable is 60,000 poundals. 

In the second place the forces c and d must be equal and opposite because the 
car has no horizontal acceleration. Let this common value of c and d he F 
poundals. 

In the third place the turning or torque action of a and h (which is 120,000 
poundal-feet) must be equal and opposite to the torque action of c and d (which 
is F X 7 poundal-feet). Therefore F X7 poundal-feet must be equal to 120,000 
poundal-feet, or F must be 17,143 poundals. 

24c. Find the value of the forces c and d in the previous 
problem (i) when the car has no acceleration, and (2) when the 
car has an upward acceleration of 6 feet per second per second. 

25a. A revolution counter shows that an electric motor makes 
1,080 revolutions while stopping (after the supply switch is 
opened), and the time which elapses is 90 seconds. Assuming 
the motor to lose speed at a constant rate find the value of this 
constant rate of loss of speed by the motor while stopping. 
Ans. 0.267 revolution per second per second, or 1.677 radians 
per second per second. 

Note. Divide total revolutions while stopping by time of stopping gives average 
speed while stopping; and this is equal to half the sum of initial speed and final 
speed (zero). Thus the initial speed is known, and it is all lost in 90 seconds so 
that average rate of loss can easily be found. 

25b. Find how far a solid cylinder will roll down a 30° incline 
in three seconds, ignoring friction. Ans. 1,471.5 centimeters, 
acceleration of gravity being 981 centimeters per second per 
second. 

Note. Draw a diagram showing an end view of the rolling cylinder, and in- 
dicate the following forces by arrows for the sake of clearness, (a) The force mg 



PROBLEMS. 535 

with which gravity pulls vertically downwards on the cylinder. This force acts 
at the axis of the cylinder. (&) The force P with which the inclined plane pushes 
on the cylinder. The down-hill component of mg is ^mg, and the component of 
mg which pulls the cylinder normally against the inclined plane is o.866wg. Let the 
up-hill component of P be represented by V, and let the component of P which 
is normal to the plane be represented by N. Then we have the following four equa- 
tions one of which is not needed for the derivation of the desired result. 

1. The net unbalanced force pulling the cylinder down hill is Img — V, and 
this force must be equal to ma, where a is the down-hill acceleration of the rolling 
cylinder. 

2. The cylinder has no acceleration in the direction normal to the plane, and 
therefore the forces which are normal to the plane balance each other, that is 
N = o.S66mg. 

3. The sum of the torque actions of all the forces about the center of mass 
(axis of figure) of the cylinder is Vr, and this torque action is equal to Ka, where K 
is the spin-inertia of the cylinder about its axis of figure ( = ^mr^ according to table 
in Art. 13), a is the spin-acceleration of the rolling cylinder, and r is the radius 
of the cylinder. 

4. During one revolution, the cylinder travels a distance 2irr, and during n 
revolutions it travels a distance 2irrn. Therefore, if the cylinder has a spin- 
velocity of n revolutions per second (2irn radians per second) its velocity will be 
2'irnr or v = sr where v is the travel velocity, and 5 is the spin- velocity of the 
rolling cylinder in radians per second. Therefore* the rate of change of v (which 
is the acceleration a of the rolling cylinder) must be r times the rate of change of 
s (which is the spin-acceleration of the rolling cylinder). 

25c. Find how far a solid sphere will roll down a 30° incline 
in 3 seconds, ignoring friction. Ans. 1576.6 centimeters, ac- 
celeration of gravity being 981 centimeters per second per second. 

25d. A 1,000-gram disk is 20 centimeters in diameter, it is 
mounted on a shaft 2 centimeters in diameter of the same 
material, and the projecting parts of the shaft have a mass of 
100 grams. The disk rolls 109.2 centimeters down an inclined 
track in 15 seconds, the drop of the track being one twentieth 
of its length along its slope. What is the acceleration of gravity? 
Ans. 980 centimeters per second per second. 

Note. The total mass of the rolling body is 10,100 grams, and its total spin- 
inertia is the sum of the spin-inertia of the complete disk and the spin-inertia of the 
projecting part of the axle. 

256. Two very fine wires are wrapped around the projecting 

* If John's capital is always k times as large as Henry's, then John's rate of 
saving or spending must always be k times as great as Henry's rate of saving or 
spending. 



536 APPENDIX A. 

ends of the axle of the disk which is described in the previous 
problem, and the disk is allowed to fall vertically downwards 
and set itself spinning as the fine wires unroll from the axle. 
Find the downward acceleration of the disk, acceleration of 
gravity being 980 centimeters per second per second. Ans. 
1940 centimeters per second per second. 

25f. What is the combined upward pull of both wires in the 
previous problem? Ans. 9,700,000 dynes or the weight of 
9,900 grams. 

26. A forward pull of 10 "pounds" is exerted at the center 
of a 500-pound wheel which is 6 feet in diameter and rolls with- 
out friction on a hard concrete floor. The spin-inertia of the 
wheel is 2,000 pound-feet-squared. Find the forward accelera- 
tion of the wheel, Ans. Forward acceleration is 0.443 feet per 
second per second. 

Note. Let B be the backward force exerted on the rim of the wheel by the 
floor (this backward force is due to the spin-inertia of the wheel not to friction). 
Then the net forward force acting on the wheel is 320 — JS poundals so that we 
have 

320 — B = 500a (i) 

where a is the unknown acceleration of the wheel. 

The torque action of the forward pull of 10 "pounds" about the axis of the 
wheel is zero, and the torque action of the force B is 3^ poundal-feet, B being 
expressed in poundals. Therefore 

ZB = 2000a (ii) 

where a is the spin-acceleration of the wheel. 

But V = rs so that a = rot, where v is the velocity and a is the acceleration 
of the wheel, and 5 is the spin- velocity and a the spin- acceleration of the wheel 
as explained in Art. 34. Therefore, using r = 3 feet, we have: 

a = 30; (iii) 

27. The forward pull of 10 "pounds" is exerted at the topmost 
point of the rim of the wheel in the previous problem. Find the 
forward acceleration of the wheel. Ans. 0.886 feet per second 
per second. 



PROBLEMS. 537 

PROBLEMS. CHAPTER II. 

28. A table top is 10 feet long and 50 inches wide. Find its 
area without reducing the data in any way. 

Note. The product of length and breadth always gives area, even when length 
and breadth are expressed in different units. 

29. A water storage basin has an area of 2,000 acres; find the 
volume of water required to fill the basin to a depth of 16 feet. 
Ans. 32,000 acre-feet. One acre foot is equal to 325,800 gallons. 

30. A man travels at a velocity of 6 feet per second. How far 
does he travel in two hours? Find the result without reducing 
the data in any way. Ans. 12 hour-feet-per-second. One 
hour X (foot per second) being the distance traveled in one 
hour by a body moving at a velocity of one foot per second. 

31. Given a northward force of 600 pounds and an eastward 
force of 400 pounds. What is the value of their resultant and 
in what direction does it act? Ans. 721 pounds acting 56° 19' 
north of east. 

32. A stream flows southwards at a velocity of 2 miles per hour, 
and a boatman rows towards the east at a velocity of 4 miles 
per hour. What is the actual velocity of the boat and in what 
direction is it traveling? Ans. 4.47 miles per hour in a direction 
63° 26' east of south. 

Note. The actual velocity of the boat is the vector sum or resultant of the 
velocity of the stream and the velocity at which the boat is rowed across the stream. 

33. A man travels 3 miles northwards and then 7 miles in a 
direction 40° east of north. How far and in what direction i? he 
from his starting point? Ans. 9.5 miles in a direction of 28° 16' 
east of north. 

34. New York City Is 82.4 miles from Philadelphia in a direc- 
tion 49° 37' east of north. How far is New York City north 
of Philadelphia and how far east? Ans. 53.4 miles north and 
62.8 miles east. 

35. A guy wire pulls on a vertical pole with a force of 200 
" pounds " and the angle between the pole and the wire is 52°. 



538 APPENDIX A. 

Find the vertical and horizontal components of the pull of the guy 
wire. Ans. Vertical component 123 " pounds," horizontal com- 
ponent 158 "pounds." 

36. A sled is pulled along a level road by a force of 37 
" pounds," the direction of the pull being 48° above the horizon- 
tal. What is the forward pull on the sled? Ans. 24.8 "pounds." 

37. A steamship is moving in a direction 42° north of east at 
a velocity of 14 miles per hour. Find the northward and east- 
ward components of its velocity. Ans. 9.37 miles per hour north- 
wards; 10.4 miles per hour eastwards. 

38. The vertical component of the pull of a guy wire is 400 
" pounds " and the angle between the guy wire and the horizontal 
is 50°. Find the total pull of the guy wire. Ans. 522 " pounds." 

39. Five hundred grams of water leak out of a pail in 25 
seconds. What is the average rate of leak? 

40. A man's wages increase from $50 per month to J140 per 
month in the course of 18 months. What is the average rate of 
increase of wages? Ans. Five dollars per month per month. 

41. The velocity of a body increases from 50 feet per second to 
140 feet per second in 18 seconds. What is the average rate of 
increase of velocity? 

42. A train gains a speed of 32 miles per hour in 80 seconds. 
Find the average rate of gain of velocity without reducing the 
data in any way. Ans. 0.4 mile per hour per second. 

43. A pole 22 feet long is dragged sidewise over a field at a 
velocity of 8 feet per second. At what rate does the pole sweep 
over area? 

44. A prism has a base of 25 square centimeters and its height 
is increasing at the rate of 5 centimeters per second. At what 
rate is the volume of the prism increasing? 

45. One side of a brick wall is at a temperature of 0° C. and 
the other side of the wall is at 23° C, and the wall is 30.5 centi- 
meters thick. What is the average temperature gradient through 
the wall? Ans. 0.754 centigrade degree per centimeter. 

46. A 200-gram ball is tied to a string and twirled in a circle 



PROBLEMS. 539 

of which the radius is 75 centimeters, and the ball makes 2 
revolutions per second, (a) What is the velocity of the ball; 
(b) What is the acceleration of the ball ; (c) What is the pull of 
the string on the ball? Ans. (a) 942.45 centimeters per second; 
(b) 11,850 centimeters per second per second towards the center 
of the circle; (c) 2,370,000 dynes. 

47. An 80-ton locomotive goes round a railway curve of which 
the radius is 600 feet at a velocity of 65 feet per second. With 
what horizontal force in "pounds" do the flanges of the loco- 
motive wheels push against the outer rail, the outer rail not being 
elevated? Ans. 32,210 "pounds." 

48. Calculate the proper elevation to be given to the outer 
rail on a railway curve 600 feet in radius for a speed of 65 feet 
per second, the width of the track being 4 feet 8| inches. Ans. 
1.015 feet. 

Note. Of course 600 feet radius is an unusually sharp railway curve. This 
problem refers strictly to a lone car or locomotive rounding the curve. A moving 
train is under tension like a belt, and its tension introduces a new condition. 

49. The stone in Fig. 36 makes 2 revolutions per second. 
What is the value of U in Fig. 36 in radians per second? The 
velocity v of the stone is, say, 942.45 centimeters per second. 
What is the value of the acceleration Av/At of the stone? The 
mass of the stone is 200 grams. What is the pull F of the string 
in dynes? Ans. (a) 12.5664 radians per second; (5) 11,850 centi- 
meters per second per second; (c) 2,370,000 dynes. 

Note. This problem is stated in this particular way in order to elucidate the 
following problem. 

The direction of v in Fig. 36 turns once around while the stone makes one 
revolution in its orbit. Therefore v turns 2 revolutions per second or 47r radians 
per second (= Q). 

The angle A<^ in radians is equal to Q-At, and the arc Av is equal to the 
product of radius (=v) and angle (^-At). Therefore Av =vQ-At, or AvjAt = vQ,. 
That is to say, the acceleration of the stone is equal to the velocity v of the stone 
multiplied by the speed of turning (fi in Fig. 37) in radians per second. 

50. The gyroscope wheel in Fig. 38 has a spin-Inertia of 500 
c.g.s. units [gram- (centimeters)^], its spin-velocity Is 100 revolu- 
tions per second (628.32 radians per second = s In Fig. 39), 



540 APPENDIX A. 

and the gyroscope wheel sweeps once around in 4 seconds (12 
in Fig. 39 equals one quarter of a revolution or 1.5708 radians 
per second), (a) What is the value of the spin-acceleration 
A5/A/ of the wheel; and (b) What is the value of the torque T 
which is causing the gyroscope wheel to swing round? Ans. 
987.2 radians per second per second, (b) 493,600 dyne-centimeters 
of torque. 

Note. The argument of this problem is precisely the same as the argument of 
the previous problem. 

The turning motion of the axis of the spinning wheel, in Fig. 38 is called preces- 
sion 

51. A side-wheel steam boat Is suddenly turned to port and 
the gyrostatic action of the paddle wheels causes the boat to 
list. Make a diagram somewhat similar to Figs. 38 and 39 and 
show which way the boat lists and why. 

Note, The starboard side of the vessel is pushed downwards. Starboard and 
port mean respectively right and left to a person facing the bow of a vessel. 

52. The flywheel of an automobile engine turns in a clock- 
wise direction as seen from in front of the car. Make a diagram 
somewhat like Figs. 38 and 39 and determine the directions of 
the forces with which the engine shaft pushes against Its front 
and back bearings (a) when the front wheels of the machine 
rise suddenly on a bump in the road, and (b) when the machine 
turns suddenly to the right. 

53. The spin-Inertia of the engine shaft and flyw^heel of a 
Ford automobile is about 30 pound- (feet) ^, The engine speed Is, 
say, 600 revolutions per minute, and the car turns in a circle 
of 25-foot radius at a speed of 30 miles per hour. Find the 
torque which must act upon the flywheel shaft to make the fly- 
wheel and shaft turn round with the car. Suppose the distance 
between front and back bearings Is 2 feet, and suppose that the 
above mentioned torque is produced by equal and opposite 
forces at the bearings. Find the value of each force. Ans. (a) 
3,317 poundal-feet ; (b) 1,655 poundals or 51.72 "pounds." 

54. The vessel described in problem 51 Is steered In a circle 
150 feet In radius at a velocity of 25 feet per second, and the 



PROBLEMS. 541 

vessel lists 5° because of the gyrostatic action of the paddle 
wheel and shaft. To produce a 5° list when the boat is standing 
still requires a weight of 10 tons (640,000 poundals) to be shifted 
from the center of the boat to a point 15 feet from the center. 
The paddle wheels make 75 revolutions per minute. Find the 
spin-inertia of the paddle wheels and shaft. Ans. 7,337,000 
pound- (feet) ^ 

55. A force of five million dynes deflects the end of the spring 
in Fig. 44 through a distance of 1.25 centimeters. What is the 
value of the constant k in equation (13) of Art. 37, and in what 
terms is k expressed? How much force would be required to 
deflect the end of the spring 2 centimeters? 

56. A mass of 2 kilograms is attached to the end of the spring 
in Fig. 44, and the mass is set vibrating. Using the data of the 
previous problem find the number of round-trip vibrations of 
the weight in one minute. Ans. 427 round-trip vibrations. 

57a. A flat spring clamped horizontally in a vise has its end 
pulled down 4.803 centimeters by a 2,000-gram body hung so as 
to pull sidewise on the end. The spring is then arranged as 
shown in Fig. 44 with the same projecting length as before and 
with the 2,000-gram body attached to the end of the spring. 
Under these conditions the body is observed to make 227 complete 
vibrations in 100 seconds. Find the acceleration of gravity 
neglecting the mass of the spring. Ans. 979 centimeters per 
second per second. 

57b. What is the length / of an ideal simple pendulum which 
makes one round-trip vibration per second at a place where the 
acceleration of gravity is 981 centimeters per second per second? 
Ans. 24.85 centimeters. 

PROBLEMS. CHAPTER HI. 

WORK AND ENERGY. 

58. A 165-pound man climbs a height of 40 feet in 1 1 seconds. 
How much work is done, and at what rate? Express the work 
in foot-" pounds," and in joules; and express the power in foot- 



542 APPENDIX A. 



It 



pounds" per second, in horse-power, and in watts. Ans. 6,600 
foot-' 'pounds" or 8,940 joules of work done at the rate of 600 
foot-' 'pounds" per second, or 1.09 horse-power, or 814 watts. 

59. A horse pulls upon a plow with a force of 100 "pounds" 
and travels 3 miles per hour. What power is developed? Ex- 
press the result in foot-" pounds" per second, in horse-power, 
and in watts. Ans. 440 foot-" pounds" per second; 0.8 horse- 
power; 597 watts. 

60. A belt traveling at a velocity of 70 feet per second trans- 
mits 360 horse-power. What is the difference in the tension of 
the belt on the tight and loose sides in "pounds"? Ans. 2,829 
"pounds." 

61. The engines of a steamship develop 20,000 horse-power, 
of which 30 per cent, is represented in the forward thrust of the 
screw in propelling the ship at a speed of 17 miles per hour. 
What is the forward thrust of the screw in "pounds"? Ans. 
132,350 "pounds." 

Note. The useful part of the power developed by the engines of a steamship 
is represented by the forward thrust of the propeller shaft against the framework 
of the ship, and the useful power is equal to the product of this force times the 
velocity of the ship. 

62. An electric motor has an efficiency of 80 per cent, and 
electrical energy costs 5 cents per kilowatt-hour. How much 
does the output of the motor cost per horse-power hour, ignoring 
interest on cost of motor and its depreciation? Ans. 4.66 cents. 

63. A 1,000 horse-power boiler and engine plant costs about 
$70,000 complete, including land, building, boilers, engines and 
auxiliary apparatus such as pumps and feed water heaters. The 
cost of operating this plant continuously, night and day, Is as 
follows : 

Interest on investment 5 per cent per annum. 

Depreciation lo 

Maintenance and repairs 4 " 

Taxes and insurance 2 ' 

Labor S30 per day, 365 days in year. 

Coal $2.00 per ton. 



PROBLEMS. 543 

The average demand for power is 50 per cent, of the rated 
power output of the plant, that is 500 horse-power, and the con- 
sumption of coal is 2J pounds per horse-power-hour. Find the 
cost of each horse-power-hour delivered by the engine. Ans. 0.83 
cent. 

64. The above engine will drive a 700 kilowatt dynamo, that is 
a dynamo capable of delivering 700 kilowatts. The cost of 
dynamo, station wiring and switch-board apparatus is $20,000. 
The average output of the dynamo is 350 kilowatts (correspond- 
ing to 500 horse-power output of engine). Calculate the cost of 
electrical energy per kilowatt-hour at the station, allowing 21 per 
cent, for interest, depreciation, etc., on the electrical machinery 
and allowing $5 per day for extra labor. Ans. i .38 cents. 

65. To determine the power developed by an electric motor a 
brake is applied to the motor pulley, and the observed brake pull 
on the periphery of the pulley is 165 " pounds." The diameter of 
the pulley is 16 inches and its speed under test is 900 revolutions 
per minute. What amount of power is developed by the motor? 
Ans. 18.84 horse-power. 

Note. The power is equal to the brake pull on the pulley rim multiplied by the 
velocity of the pulley rim. 

66. A steamship has a mass of 25,000 tons. What is the 
kinetic energy of the ship at a speed of 18 miles per hour? Ex- 
press the result in foot-" pounds" and in horse-power-hours. 
Ans. 544.5 X 10^ foot-" pounds " ; 275 horse-power-hours. 

Note. Take one ton equal to 2,000 pounds. 

67. A bicycle rider has a 50-foot hill to climb. What velocity 
must he have at starting to relieve him from the doing of one 
third of the work required? Ans. 32.7 feet per second. 

68. The rim of the flywheel of a metal punch press has a 
mass of 560 pounds. What must be the initial velocity of the 
rim in feet per second in order that the press may exert a force 
of 72,000 "pounds" while moving a distance of one inch, and 
reduce the velocity of the rim to 70 per cent, of its Initial value? 



544 



APPENDIX A. 



Assume the entire rim to have the same velocity and ignore the 
kinetic energy of moving spokes and hub. Ans. 36.66 feet per 
second. 



Note. Let v be the initial velocity of the rim and v' its final velocity. Then 
its initial kinetic energy is ^mv^ foot-poundals, and its final kinetic energy is 
fm/^ foot-poundals. The work done by the press is 72,000 "pounds" X x\ foot 
or 6,000 foot-" pounds" or 192,000 foot-poundals. Therefore \mv^ — ^mv'"^ 
= 192,000 and v' = o.']v. 



PROBLEMS. CHAPTER IV. 

HYDROSTATICS. 

69. The piston of a steam engine is 24 inches in diameter and 
the steam pressure is 150 "pounds" per square inch. Find the 
force with which the steam pushes on the piston. 

70. Calculate the circumferential tension in the shell of a 
boiler, the diameter of the boiler being 6 feet and the steam 
pressure 125 "pounds" per square inch. 

71. A given grade of steel can stand safely a tension of 20,000 
"pounds" per square inch. What is the greatest diameter of 
steel tube with walls 0.02 inch thick which can safely withstand 
a pressure of 150 "pounds" per square inch? Ans. 5.33 inches 
in diameter. 

72. The reservoir of a city water supply is 210 feet above a 
certain hydrant. What is the water pressure at the hydrant 
when the water in the city mains is standing still? Ans. 13,125 
"pounds" per square foot or 91. i "pounds" per square inch. 

73. What air pressure is required to hold the water level in 
a caisson 62 feet below the water level in a river? 

74. The density of mercury at 0° C. is 13.5956 grams per 
cubic centimeter. Calculate the value in dynes per square centi- 
meter of standard atmospheric pressure, namely 76 cm. of mer- 
cury at 0° C, the value of gravity being 980.61 cm. per second 
per second. Ans. 1,012,900 dynes per square centimeter. 

75. The specific gravity of mercury is approximately 13.6. 
The pressure in "pounds" per square inch at a point x feet 
beneath pure water is ^ = 0.434X. Find the value in pounds per 



PROBLEMS. 545 

square inch of one English standard atmosphere, namely, 30 
inches of mercury. Ans. 14.75 "pounds" per square inch. 

76. Calculate the height of the homogeneous atmosphere; that 
is, assuming that the atmosphere has a uniform density of 
0.00129 gram per cubic centimeter throughout, calculate the 
depth which would produce standard atmospheric pressure. 
Ans. 8,012 meters or 4.98 miles. 

Note. Take 980 centimeters per second per second for the acceleration of 
gravity. 

77. A piece of lead weighs 233.60 grams in air and 212.9 
grams in water at 20° C. What is the specific gravity and the 
density of lead at 20"^' C? Ans. Specific gravity 11.285; density 
11.265 grams per cubic centimeter. 

Note. The density of water at 20° C. is 0.998252 gram per cubic centimeter. 

78. A piece of glass weighs 260.7 grams in air and 153.8 
grams in water at 20° C. The same piece of glass weighs 92.2 
grams in dilute H2SO4 at 20° C. What is the specific gravity of 
the H2SO4 at 20° C? Ans. 1.576. 

79. A glass bulb weighs 75.405 grams when filled with air at 
standard temperature and pressure. It weighs 74.309 grams 
when the air is pumped out. It weighs 74.385 grams when 
filled with hydrogen at the same temperature and pressure. 
What is the specific gravity of hydrogen referred to air? Ans. 
0.06926. 

80. What is the net lifting capacity of a balloon containing 
400 cubic meters of hydrogen, its material weighing 250 kilo- 
grams? (Weight of a cubic meter of air is 1,200 grams; weight 
of a cubic meter of hydrogen is 90 grams.) Ans. 194 kilograms. 

PROBLEMS. CHAPTER V. 

HYDRAULICS. 

81. Find the mean velocity at which water must flow in a 
canal 20 feet wide and 6 feet deep, in order that the rate of dis- 
charge may be 500 cubic feet per second. 

36 



546 APPENDIX A. 

82. The velocity of the water in a pipe is 2.3 feet per second 
where the inside diameter of the pipe is 8 inches; what is the 
velocity of the water where the inside diameter of the pipe is 
5 inches? 

83. How much work is required to pump 6,750 cubic centi- 
meters of water (assumed to be incompressible) into a tank in 
which the pressure is 24 X 10^ dynes per square centimeter? 
Ans. 162 X 10^ ergs. 

Note. The dyne and the erg are inconveniently small units for many practical 
purposes. 

84. How much work is required to pump six cubic feet of 
water (assumed to be incompressible) into a tank where the 
pressure is 100 "pounds" per square inch? Ans. 86,400 foot- 
-pounds." 

Note. It is most convenient to use c.g.s. units or f.p.s. units in equations 31-36. 
See Art. 18. 

85. What is the kinetic energy per cubic foot of water which 
is moving at a velocity of 200 feet per second? Ans. 390,625 
foot-" pounds." 

86. Calculate the velocity of efflux of kerosene from a tank 
in which the pressure is 50 "pounds" per square inch above 
atmospheric pressure, the density of kerosene being, say, 48 
pounds per cubic foot. Ans. 98 feet per second. 

87. Water flows in a 12-inch main at a velocity of 4 feet per 
second and encounters a partly closed gate valve through which 
the stream of water is reduced to 0.36 square foot section. 
Calculate the loss of pressure at the valve due to friction. Ans. 
0.408 "pounds" per square inch. 

Note. As the water enters the narrow passageway through the valve its velocity 
increases and its pressure drops off accordingly, as explained in Art. 67. But as the 
water issues from the narrow passageway it retains its velocity as a jet flowing 
through the surrounding water so that its pressure does not rise again — the excess 
velocity is destroyed by eddy action. Therefore the loss of pressure through the 
valve is approximately equal to the drop of pressure due to the increased velocity 
of the water as it enters the valve. 

88. A street water-main 7 inches inside diameter has in it a 



PROBLEMS. 547 

throat 3 Inches inside diameter. Water flows through the pipe 
at the rate of 1.5 cubic feet per second and the pressure of the 
water in the 7-inch pipe is 90 "pounds" per square inch. What 
is the pressure in the throat, ignoring friction? Ans. 84 "pounds" 
per square Inch. 

89. A fire-test is made to determine the friction loss of pres- 
sure in a street main when a fire hose is in action, and it is ob- 
served that the pressure at the hydrant drops from 150 "pounds" 
per square inch to 90 "pounds" per square Inch when 6 cubic 
feet of water per second is being discharged, the sectional area of 
the pipe at the point of attachment of the pressure gaug^ being 
0.0833 square foot. What Is the friction loss of pressure be- 
tween the reservoir and the point of attachment of the pressure 
gauge? Ans. 24.8 "pounds" per square inch. 

Note. It might seem that the requked friction loss of pressure is 150 "pounds" 
per square inch minus 90 "pounds" per square inch. But there is a drop in 
pressure due to the fact that the water is moving in the pipe to which the pressure 
gauge is attached. It is assumed that the pressure gauge indicates the actual 
pressure in the pipe to which it is connected. 

90. A pair of Pitot tubes Is placed In a river as shown in Fig. 
85 and the difference of level h Is observed to be 3 inches. What 
is the velocity of the water in the river at the place where the 
meter Is placed? Ans. 4 feet per second. 

91. A pair of Pitot tubes is placed In an air blast as shown in 
Fig. 86, and the difference of level h Is observed to be 2 J inches, 
the liquid in the tube being water. What is the velocity of the 
air blast? Ans. 101.2 feet per second. 

Note. In this problem ignore the compressibility of the air. Take the density 
of the air to be 0.0013 of the density of water. 

92. How much water will flow per second through 1,000 feet 
of I -Inch pipe with an available head of 10 feet all to be used 
in pushing the water through the pipe. Ans. 0.00879 cubic foot 
per second. 

Note. A part of the available head is used to set the water in motion. This 
part, expressed in poundals per square foot, is equal to ^d'J^, where d is the 
density of water in pounds per cubic foot and v is the velocity of the water in the 
pipe in feet per second. This detail has been ignored in obtaining the above answer. 



548 APPENDIX A. 

PROBLEMS. CHAPTER VI. 

TEMPERATURE AND THERMAL EXPANSION. 

93. The cubic contents of an oxygen tank is 3.5 cubic feet 
and oxygen is stored in the tank at a pressure of 200 "pounds" 
per square inch. How many cubic feet of oxygen is there in the 
tank reckoned at atmospheric pressure (15 "pounds" per square 
inch) and at the same temperature? 

94. The air in the bulb of an air thermometer has a pressure 
of 750 milHmeters when the bulb is placed in a steam bath (at 
standard atmospheric pressure) and a pressure of i ,203 millimeters 
when the bulb is placed in a bath of lead at its melting point. 
What is the temperature of melting lead reckoned from ice 
point? Ans. 325° C. 

Note. In solving this problem ignore the slight increase of volume of the bulb 
between steam temperature and the temperature of melting lead. 

95. The pressure of the air in an air thermometer bulb at 
steam temperature (at standard atmospheric pressure) is 1.367 
as great as the pressure of the air in the bulb when the bulb is in ice 
water. The difference between ice temperature and steam tem- 
perature is taken arbitrarily to be 180 degrees on the Fahrenheit 
scale. Find the Kelvin temperature of freezing water in Fahren- 
heit degrees. Ans. 491 degrees. 

96. {a) Reduce to Fahrenheit the following centigrade tem- 
peratures: 45° C, 12° C. and — 20° C. {h) Reduce to centigrade 
the following Fahrenheit temperatures: 212° F., 72° F., 32° F. 
and — 30° F. 

97. The stem of a thermometer has upon it a scale of equal 
parts, and the ice point and the steam point of the thermometer 
are observed to be at a distance of 92.6 of these divisions apart. 
(a) At what point will the mercury stand at a temperature of 
67° C? {h) At what point will the mercury stand at a tempera- 
ture of 120° F.? Ans. (a) 62.04. (&) 45.3. 

98. Suppose a thermometer stem to be divided into 140 equal 
spaces between the ice point and the steam point and suppose 



PROBLEMS. 549 

the marks to be numbered upwards from the tenth division below 
the ice point, making the ice point No. lo. Reduce the following 
readings of this thermometer to centigrade and to Fahrenheit: 
150°, 70°, 0°, and - 20°. Ans. 107°.! C; 42°. 8 C; - 7°.! C; 
- 2i°.4 C; 224°.9 F.; 109°.! F.; 19°.! F.; - 6°.6 F. 

99. An amount of gas at 15° C. has a volume of 120 c.c. Find 
its volume at 87° C, the pressure being unchanged. Ans. 150 
cubic centimeters. 

100. A volume of hydrogen at 11° C. measures 4 liters. The 
gas is heated until its volume is increased to 5 liters without 
changing the pressure. Find the new temperature. Ans". 82° C. 

loi. A flask containing air at 760 millimeters pressure is corked 
at 20° C. Find the pressure of the air in the flask after it has 
stood in a steam bath at 98° C, neglecting the slight increase 
of volume of the flask. Ans. 962 millimeters. 

102. The density of dry air at 0° C. and 760 millimeters is 
0.001293 gram per cubic centimeter. What is the volume of 
25 grams of air at 25° C. and at a pressure of 730 millimeters? 
Ans. 21,970 cubic centimeters. 

Note. When the pressure, volume and temperature of a gas all change, calcu- 
lations can be mo^t easily made by using the equation 

pv _ p'v^ 

This form of equation obviates any consideration of the values of the constants 
M and R in equation (46). 

103. A quantity of gas is collected over mercury in a eudiom- 
eter tube. The volume of the gas is observed to be 50 cubic 
centimeters, its temperature is 10° C, the level of the mercury 
in the tube is 10 centimeters above the level of the mercury in 
the basin, and a barometer shows that the atmospheric pressure 
is 750 millimeters. Find the volume the gas would occupy at 
0° C. and 760 millimeters. Ans. 41.25 cubic centimeters. 

104. A steel meter scale is 99.981 centimeters long at 10° C. and 
100.015 centimeters long at 40° C. At what temperature will 
the scale be exactly one meter long, assuming the expansion from 



550 APPENDIX A. 

10° C. to 40° C. to be proportional to the increase of temperature? 
Ans. 26°.8 C. 

105. A piece of soft wrought iron was found by Andrews to 
have a length of 101.5 centimeters at a temperature of 100° C. 
and a length of 101.77 centimeters at a temperature of 300° C. 
Find the mean coefficient of linear expansion of the iron between 
100° C. and 300° C. Ans. 0.0000133. 

Note. The mean coefficient of linear expansion between two temperatures is 
defined as the difference in length at the two temperatures divided by the length 
at the lower temperature and by the difference of temperature. From this defi- 
nition we have 

Lt' = Lt[i + a{t' - t)\ 

The value of a in this equation is slightly different from the value of a in 

the equation 

Lt = Lo(i + at) 

but the difference is very small and is entirely negligible when one uses a tabulated 
value of a coefficient of linear expansion for purposes of calculation, unless the 
metal to which the calculation applies is known to be identically the same kind of 
metal as that for which the tabulated value of the coefficient was determined. 

106. An iron steam-pipe is 1,000 feet long at 0° C. and it ranges 
in temperature from — 20° C. to 115° C. What must be the 
range of motion of an expansion joint to provide for expansion? 
Ans. 1.539 feet. 

Note. The coefficient of linear expansion of wrought iron for the given range 
of temperature is 0.0000114 according to Andrews. 

107. A surveyor's steel tape is correct at 0° C. A distance as 
measured by the tape at 22° C. is 500 feet. What is the true 
value of the measured distance, coefficient of linear expansion of 
steel being o.ooooi II? Ans. 500.1221 feet. 

Note. The measured distance is 500 times as long as the portion of the tape 
between two adjacent foot-marks at the temperature at which the tape is used. 

108. Assuming extreme range of temperature from — 15° C. 
to 45° C, find the range of expansion of one of the 1,700-foot 
spans of the Forth Bridge. The bridge is made of steel of 
which the coefficient of linear expansion is about 0.00001 13 per 
degree centigrade. 



PROBLEMS. 551 

109. A steel shaft is 20 inches in diameter at 70° F. A steel 
collar is to be shrunk upon this shaft. The collar is to be heated 
to 650° F. and have at that temperature an inside diameter of 
20.01 inches, so that it may be easily slipped over the shaft. 
Required the inside diameter to which the collar must be turned 
in the shop, shop temperature being 70° F. The coefficient of 
linear expansion of steel is 0.0000113 per degree Centigrade. 
Ans. 19.937 inches. 

no. A glass bottle is weighed as follows: (a) empty, 24.608 
grams; (b) full of mercury at 0° C, 258.723 grams; and (c) full 
of mercury at 100° C, 255.133 grams. Find the coefficient of 
cubic expansion of the glass of which the bottle is made. Ans. 
0.000027. 

Note. The space inside of a vessel increases exactly as if it were a solid piece 
of the material of which the vessel is made. The density of mercury at o° C. is 
13-5956 grams per cubic centimeter, and at ioo° C. it is 13.3524 grams per cubic 
centimeter. 

PROBLEMS. CHAPTER VII. 

CALORIMETRY . 

111. How many watts are required to raise the temperature of 
5,000 grams of water from 10° C. to 35° C. in 20 minutes? Ans. 
436.27 watts. 

Note. This problem is to be solved accurately by the use of the table in Art. 82. 
In most practical calculations, however, the specific heat of water may be considered 
as constant and equal to 4.2 joules per gram per degree centigrade. 

112. How much does it cost raise to 5,000 grams of water 
from 0° C. to 100° C. by an electric heater when electrical 
energy costs 10 cents per kilowatt- hour and 20 per cent of the 
heat is wasted? Ans 5.83 cents. 

Note. See problem 117. 

113. Niagara Falls is 165 feet high. Calculate the rise of tem- 
perature of the water due to the energy of the fall on the assump- 
tion that the cooling effect of vaporization is zero. Ans. 0.212° F. 

114. A copper calorimeter weighing 50 grams contains 500 



552 APPENDIX A. 

grams of water at i6° C. A piece of copper weighing 65 grams 
is heated to 100° C. and plunged into the water of the calorimeter. 
The resulting temperature is 17° C. Find the specific heat of 
copper. Ans. 0.0935. 

115. A piece of lead weighing 1,500 grams at 100° C. is placed 
in a vessel containing 400 grams of water at 15° C, and the 
temperature of the mixture finally settles to 24°.oi C. The 
vessel is made of copper and it weighs 50 grams. What is the 
specific heat of lead, the specific heat of copper being 0.093? 
Ans. 0.032. 

Note. The amount of heat given off by the lead is 1500/(100—24.01) calories 
where I is the specific heat of lead. The amount of heat absorbed by the water is 
400(24.01 —15) calories. The amount of heat absorbed by the vessel is 50X0.0935 
X (24.01 —15). And the heat given off by the cooling lead is equal to the total 
quantity of lead absorbed by water and vessel. 

116. A copper vessel weighing 2 kilograms contains 24 kilo- 
grams of water at 20° C. Into this vessel are dropped at the 
same instant, 2 kilograms of copper at 100° C, 2.4 kilograms of 
zinc at 60° C, and 6.4 kilograms of lead at 50° C. Find the re- 
sultant temperature. The specific heat of zinc is 0.093. Ans. 
2I°.2 C. 

Note. When there is no question as to the freezing of a portion of the water 
or the boiling of a portion of the water, the simplest argument of a problem of this 
kind is as follows: Let t be the resultant temperature and for the sake of argument 
let us think of t as being higher than any of the given temperatures. Then the 
specific heat of any one of the given substances multiplied by its mass and multiplied 
by {t minus initial temperature of substance) is the amount of heat required to 
raise the substance up to the resultant temperature. Adding all such products 
together gives the total heat required to raise the mixture up to its resultant 
temperature and this total heat is equal to zero, under the conditions of the problem. 

117. Coal giving 7,770 calories per gram and costing $6 per 
1,000 kilograms is used in a cook stove and about 5 per cent of 
the heat of combustion is utilized, for example, in heating water. 
Find the cost of heating 5,000 grams of water from 0° C. to 100° C. 
making no allowance for wear and tear on stove and cooking 
utensils. Ans. 0.772 cent. 

Note. See problem 112. 



PROBLEMS. 553 

ii8. The heat of combustion of hydrogen is 34,000 calories 
per gram. How many calories does this represent per gram of 
oxygen, and how many calories per gram of water produced? 
Ans. 4,250 calories per gram of oxygen, 3,777 calories per gram 
of water. 

119. The heat of combustion of pure charcoal is 4,000 British 
thermal units per pound when the product of the combustion 
is carbon monoxide (CO) and 14,500 British thermal units per 
pound when the product of the combustion is carbon dioxide 
(CO2). What is the heat of combustion of carbon monoxide 
(CO) per pound? Ans. 4,500 British thermal units per pound. 

120. A boiler shell is made of iron one centimeter thick, and 
20,000 calories per hour flow through each square centimeter of 
the shell. What is the temperature difference between inner and 
outer surfaces of shell? What would this temperature difference 
be with a copper shell of the same thickness? Use values of 
thermal conductivity given in Art. 86. Ans. 34°. 7 C. ; 5°. 78 C. 

121. The inside surface of the window-glass in a house is at a 
temperature of 15° C, the outside surface is at a temperature of 
— 10° C, and the glass is 4 millimeters thick. Calculate the heat 
in calories which during 24 hours flows out of the house by con- 
duction through a total window-glass area of 20 square meters 
and reduce the result to kilograms of coal at 6,000 calories per 
gram. Use the value of thermal conductivity given in Art. 86. 
Ans. 1,728 X 10^ calories; 288 kilograms of coal. 

122. A metal vessel containing 25,200 grams of water has a flat 
face 30 X 30 centimeters which is pressed against the outside of 
a furnace wall made of brick, and the temperature of the water is 
observed to rise 7°. 5 C. in twenty minutes. The wall is 30 centi- 
meters thick, the inner face of the wall is at a temperature of 
1,500° C. and the outer face of the wall is at 25° C. What is the 
thermal conductivity of the material of which the wall is made? 
Ans. 0.00356. 



554 



APPENDIX A. 



PROBLEMS. CHAPTER VIII. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS AND GASES. 

123. An Open vessel contains 500 grams of ice at a temperature 
of — 20° C. and heat is imparted to the vessel at the rate of 10 
calories per minute. Plot a curve showing elapsed times as ab- 
scissas and temperatures of vessel as ordinates, assuming that 
the vessel gives no heat to surrounding bodies. The specific 
heat of ice is 0.51 and latent heat of fusion of ice is 80 calories per 
gram; the specific heat of steam is 0.38 and the latent heat of 
vaporization of steam at 100° C. and normal atmosphere pres- 
sure is 537 calories per gram. 

Note. See Art. 95. 

124. Find the amount of heat required to raise 3 kilograms of 
lead at 10° to its melting point and melt it. The mean specific 
heat of lead between 10° C. and its melting point (325° C.) is 
about 0.035 ^i^d the latent heat of fusion of lead is 5.9 calories 
per gram. Ans. 50,775 calories. 

125. How much water at 50° C. is required to melt 5 kilograms 
of ice at — 10° C? Ans. 8.51 kilograms. 

126. A gas is collected over water at a temperature of 15° C. 
and the observed pressure is 752 millimeters. What would the 
pressure of the given amount of gas be if it occupied the same 
volume dry, that is, free from admixture of water vapor? Ans. 
739.33 millimeters. 

127. A closed bottle is full of dry air at 720 millimeters pressure 
and at a temperature of 50° C. A small quantity of water is 
introduced into the bottle and the whole is allowed to stand 
until the water vapor is saturated throughout the enclosed space. 
What is the total pressure of air and water vapor? Ans. 811.98 
millimeters. 



PROBLEMS. 555 

PROBLEMS. CHAPTER IX. 

THE SECOND LAW OF THERMODYNAMICS. 

128. Find the amount of thermodynamic degeneration (in- 
crease of entropy) involved in the conversion of 50 kilowatt- 
hours of electrical energy into heat at 27° C. Ans. 600,000 
joules per degree. 

129. Find the amount of thermodynamic degeneration (in- 
crease of entropy) involved in the direct transfer of 1,000 calories 
of heat from 100° C. to 0° C. Ans. 0.98 calories per degree. 

130. A steam engine works between a boiler temperature of 
184° (about 150 "pounds" per square inch steam pressure) and 
a condenser temperature of 60° C, and the engine converts 13 
per cent of the heat of the steam into useful work. What is 
the thermodynamic efficiency of the engine? Ans. 47.9 per cent. 

Note. The thermodynamic efficiency of an engine is defined as ajh, where a 
is the actual efficiency (the fraction of Hi which is converted into useful work) 
and h is the efficiency of a perfect engine working between the same temperatures. 

131. The fire temperature under a boiler is 1,600° C, the boiler 
temperature is 184° C. and the condenser temperature is 60° C. 
(a) What would be the efficiency of a perfect engine working 
between fire temperature and condenser temperature? {h) What 
would be the efficiency of a perfect engine working between 
boiler temperature and condenser temperature? (c) What is the 
loss of possible efficiency due to the direct flow of heat from the 
fire to the boiler? Ans. (a) 82.2 per cent; {h) 27.1 per cent; (c) 
the difference between 82.2 and 27.1. 

132. A good gas or gasolene engine has an actual efficiency of 
30 per cent. Taking temperatures of gases immediately after 
the explosion and after the expansion as 1,500° C. and 800° C. 
find the thermodynamic effic'ency of the engine. Ans. 75.9 
per cent. 

Note. See note to problem 130. 

133. An ammonia refrigerating machine takes in heat from a 
cool room at 0° C. and gives out heat to the warm outside air at 



556 



APPENDIX A. 



35° C. If there were no friction losses nor any sweeping proc- 
esses involved in the operation of the refrigerating machine we 
would have 

where Hi is the heat given out to the warm region at Kelvin 
temperature Ti and H2 is the heat taken in from the cool 
region at Kelvin temperature T^. Find the power in watts re- 
quired to furnish an amount of refrigeration equivalent to the use 
of 25 kilograms of ice (nearly 50 pounds of ice) per day {a) on the 
assumption that the refrigerating machine is "perfect" as above 
specified, and {h) on the assumption that the machine takes 
three times as much power as a perfect machine would take. 
Ans. (a) 12.3 watts; ih) 36.9 watts. 

134. Choosing ice water as the standard or zero state, find the entropy of lOO 
grams of ice at o° C. Ans. minus 123.0 joules per degree. 

135. Assuming the specific heat of water to be constant and equal to 4.2 joules 
per gram per degree, find the entropy of W grams of water at 100° C. Ans. 
19.34 W joules per degree. 

136. The latent heat of vaporization of steam at 100° C. is 539 calories per gram. 
Find the entropy of one gram of steam at 100° C. Ans. 25.38 joules per degree. 



PROBLEMS. CHAPTER X. 

THE ELECTRIC CIRCUIT. 

137. The accompanying diagram Fig. 463 shows a lamp L with 
connections arranged so that the lamp can be turned on or off 
at switch A (or B) regardless of how switch B (or A) stands. 
Make four diagrams like Fig. 463 showing the four possible 
combinations of switch-positions, and indicate the flow of 
current, if any, by arrows. 




supply mains 



■©— -^ 



Fig. 463. 



PROBLEMS. 



557 



138. The six small circles in Fig. 464 represent the contact 
posts on a double-pole double-throw switch, and the dotted lines 
represent the switch blades. The diagram shows the lamp L 
taking current from the direct-current mains. Make a diagram 
showing the lamp taking current from the alternating current 
mains. 



A C 
supply mains 



O 



1- 



-O- 



-O 



? 



— -o- 



DC 
supply mains 



Fig. 464. 

139. The six small circles in Fig. 465 represent the contact posts 
on a double-pole double-throw switch with crossed connections 
adapting it for use as a reversing switch, and the dotted lines 
represent the switch blades. Make a diagram showing a re- 
versed flow of current through the receiving circuit R. 



M 





AO 
AO- 



r\^"^-vir 



■c 



\H 



.6 

B 



b 

B 



Fig. 465. 



Fig. 466. 



140. Figure 466 shows the diagram of connections of an 
ordinary telegraph relay, a "local" circuit connected to the 
binding posts BB is opened and closed as the lever L of the 
relay is moved back and forth by pulses of current coming 
over the telegraph line which is connected through the binding 
posts A A to ground; ikf is a screw with a metal tip, and H is 
a screw with a hard-rubber insulating tip. Make a diagram 
showing M and H interchanged, and showing A A and BB 
connected to each other and to a battery so that the relay will 
buzz like an ordinary interrupter bell. 



558 APPENDIX A. 

141. It is possible to connect any number of bell circuits to 
one battery and have any number of push buttons arranged to 
close each bell circuit, only, of course, a small battery cannot 
operate more than one or two bells simultaneously. Make a 
diagram showing two bells connected to one battery with two 
push buttons for closing the circuit of each bell. 

PROBLEMS. CHAPTER XI. 

MAGNETISM AND THE MAGNETIC EFFECT OF THE ELECTRIC 

CURRENT. 

142. In what direction did the compass needle point in 1905 
at a place 30° west of Greenwich and 40° north latitude? At a 
place 150° west of Greenwich and 70° north latitude? 

143. Two permanent magnets i centimeter X | centimeter 
X 30 centimeters long are magnetized to an intensity of 700 units 
pole per square centimeter of sectional area, (a) Calculate the 
strength of each pole, (b) Calculate the force with which the 
north pole of one rod attracts the south pole of the other rod 
when the poles are at an approximate distance of 10 centimeters 
from each other. Ans. (a) 350 units pole, (b) 1,225 dynes. 

Note. In this and the succeeding problems assume the poles of the magnet to be 
concentrated at the center of the ends of the bars. The intensity of magnetization 
of an iron rod is the strength of pole on one end divided by the sectional area of 
the rod. 

144. The two magnets specified in the previous problem are 
arranged as shown in Fig. 467. Find the total force with which 
one magnet acts upon the other. Ans. 227.39 dynes (attraction). 

s ir S N 



^^30 cm: ^'^V cm:"^ 'So'ml'''^'^ 

Fig. 467- 

145. The intensity of the earth's magnetic field at Washington 
is 0.58 gauss and its dip is 62°. Find its horizontal and vertical 
components. Ans. H = 0.272 gauss, V = 0.512 gauss. 



PROBLEMS. 559 

146. One of the magnets specified in problem 143 is balanced 
horizontally on a knife edge at Washington. The magnet weighs 
120 grams. Find the horizontal distance from the knife edge to 
the center of the bar, taking the acceleration of gravity to be 980 
centimeters per second per second. Use the data specified in 
problem 145. Ans. 0.046 centimeter. 

147. Find the intensity of the magnetic field at a point 18 
centimeters from the north pole and 24 centimeters from the 
south pole of one of the magnets of problem 143. Ans. 1.24 
gausses. 

148. A room 6 meters long by 5 meters wide by 3 meters high 
has its longest dimension magnetic north and south. The inten- 
sity of the earth's field in the room is 0.62 gauss and the dip is 
72°. Find the number of lines of magnetic flux across each of 
the walls, the ceiling, and floor of the room and specify in each 
case whether the flux is passing out of the room or into the room. 
Ans. East wall, o; west wall, o; north wall, 28,740 maxwells out; 
south wall, 28,740 maxwells in; ceiling, 176,900 maxwells in; 
floor, 176,900 maxwells out. 

149. The pole face of the field magnet of a dynamo has an area 
20 centimeters by 30 centimeters. The magnetic field between 
the pole faces and the armature core is perpendicular to the pole 
face at each point and its intensity is 6,000 gausses. Calculate 
the number of lines of force which pass from the pole face into 
the armature core. Ans. 3,600,000 maxwells. 

Note. See Figs. 141 and 142. 

150. Calculate the number of lines of force which emanate from 
the north pole of one of the magnets specified in problem 143. 
Ans. 4,400 maxwells. 

151. Figure 468 shows a magnet NS placed near a long 
straight electric wire. The wire exerts forces on the magnet 
poles iV and 6* as indicated by the arrows F' and F" . Draw 
a diagram showing the total, or resultant, force exerted on the 
magnet by the wire. 



560 



APPENDIX A. 



, Note. A north pole near a long straight electric wire is acted on by a force 
(see arrow F' in Fig. 468) which is in a plane at right angles to the wire (the plane 
of the paper in Fig. 468), the force is at right angles to a line drawn from the wire 
to the pole (the line r in Fig. 468), and the direction of the force is such that the 
pole tends to travel around the wire in the direction in which a right-handed screw 
would have to be turned to make it travel in the direction of flow of current through 
the wire. 

The force exerted on a south pole is opposite in direction to the force which 
would be exerted on a north pole at the same place. 

The magnitude of the force exerted on a magnet pole by a long straight electric 
wire is inversely proportional to the distance of the pole from the wire. 

152. The current in Fig. 468 is reversed so as to flow towards 
the reader. Make a diagram showing the forces exerted on the 
poles N and 6" by the wire, and make a diagram showing the 
total, or resultant, force exerted on the magnet. 



© wire' 




n 



N 



Fig. 468. 



Fig. 469. 



153. The small circle with a dot in Fig. 469 represents a straight 
wire at right angles to the paper with current flowing towards 
the reader. Draw arrows showing the directions and approxi- 
mate values of the forces exerted by the electric wire on the 
poles N and 5 of the magnet. 

154. Specify the direction of the side push exerted on the wire 
by the magnet pole in Fig. 470. 

Note. The force exerted on a wire by a magnetic field is at right angles to the 
wire and at right angles to the lines of force of the field, as stated in Art. 138. The 
direction of a magnetic field at a point (the direction of the lines of force at the 
point) is the direction in which a compass needle would point if placed at that 
point, the north pole of the needle being thought of as the pointing end of the 
needle. Now a magnetic needle put in place of the wire in Fig. 470 would point 
towards S, as shown by the short arrows ff in Fig. 471. Therefore, according 
to Art. 138, the wire will be pushed towards A or towards B', to determine which, 
the following considerations are sufficient: The lines of force of the magnetic field 
due to the current in the wire encircle the wire as explained in Art. 139 and as indi- 



PROBLEMS. 



56i 



cated by the curled arrows c in Fig. 471, and the heads of the curled arrows are 
in the direction in which a right-handed screw would have to be turned to travel 
in the direction of flow of the current in the wire. Now the lines of force ff bend 
to one side of the wire so as to go with the curled arrows c, as shown in Fig. 472; 
and the tension of these bent lines of force pushes sidewise on the wire in the 
direction of the arrow F as explained in Art. 140. 




©irire 



E 



A 

k 



f^ 



Fig. 470. 



B 

Fig. 471. 





wir^ 



Fig. 472. 



Fig. 473- 



155. Specify the direction of the side push exerted on the wire 
by the magnet pole in Fig. 473. 

156. A horizontal wire 10 meters long, stretched due magnetic 
east and west, is pushed up by the horizontal component of the 
earth's field with a force of 2,500 dynes. What is the direction 
and strength of the current in the wire? The horizontal com- 
ponent of the earth's field is 0.2 gauss. Ans. 125 amperes east. 

157. The armature of a dynamo has a length, under the pole- 
face, of 30 cm. The magnetic field intensity between the pole- 
face and the armature core is 6,000 gausses. The surface of the 
armature is covered with straight wires parallel to the axis of the 
armature. Each of these wires carries a current of 75 amperes. 
Calculate the force acting on each wire. Ans. 1,350,000 dynes. 

158. A horizontal electric light wire stretched due magnetic 
north and south carries i ,000 amperes of current flowing towards 
the north. The length of the wire is 250 meters, the intensity 
of the earth's field is 0.57 gauss and the magnetic dip is 63°. 
Find the value of the force pushing on the wire and specify its 
direction. Ans. 1,269,500 dynes west. 

159. A rectangular frame 25 X 40 cm. has 10 turns of wire 
wound upon it. The frame is balanced horizontally upon an axis 

37 



562 APPENDIX A. 

pointing due magnetic east and west. A current of 28 amperes 
is sent through the wire. Required the distance from the axis at 
which a lo-gram (9,800-dyne) weight must be hung to balance 
the torque action due to the earth's magnetic field at a place 
where its intensity is 0.57 gauss and its dip is 63°. Ans. 0.74 
cm. 

160. A circular coil of wire of 20 cm. radius has 15 turns of 
wire. How much current is required in the coil to produce at 
the center of the coil a field intensity of 0.57 gauss? Ans. 0.121 
abamperes. 

161. A tangent galvanometer gives a deflection of 10° for 1.2 
amperes. Calculate the deflection which will be produced by 15 
amperes. Ans. 65° 35'. 

PROBLEMS. CHAPTER XH. 

THE CHEMICAL EFFECT OF THE ELECTRIC CURRENT. 

162. The anode of an electrolytic cell is a copper rod one inch 
in diameter, and the cathode is a hollow copper cylinder 6 inches 
in diameter. The two electrodes are co-axial and they stand 
on a flat glass plate in an electrolyte which is 8 inches deep over 
the glass plate. A current of 5 amperes flows through the cell. 
Find the current density at the cathode and the current density 
at the anode. Ans. 0.033 ampere per square inch at the cathode, 
and 0.199 ampere per square inch at the anode. 

Note. The current density on an electrode is the current per unit of area, and 
it is generally non-uniform. In the conditions specified in the problem, however, 
the current density is uniform over the surface of each electrode. Current density 
at an electrode is important inasmuch as the character of the chemical action at 
an electrode depends in many cases on current density, and the 'physical character 
of a deposited metal depends on the current density. 

163. How long a time In seconds would be required for one 
ampere to deposit one gram-equivalent (107.93 grams) of silver 
from a solution of pure silver nitrate? Ans. 26.82 hours. 

164. The formula for copper nitrate (cupric) Is Cu(N03)2, and 
the formula for silver nitrate is AgNOs. // is found hy experi- 
ment that the same amount of nitric acid radical (NO3) is set free 



PROBLEMS. 563 

at the anode per ampere per second from any nitrate solution. 
Knowing that 0.001118 gram of silver is deposited per second 
by one ampere, find how much copper is deposited per second 
by a current of one ampere from a solution of cupric nitrate. 
Ans. 0.000329 gram. 

165. The formula for cuprous nitrate is CuNOs. From the 
data of the previous problem find the amount of copper which 
would be deposited by one ampere in one second from a solution 
of cuprous nitrate. Ans. 0.000658 gram. 

166. A current which gives a steady reading of 10 "amperes'* 
on an ammeter is found to deposit 8.24 grams of copper in 40 
minutes from a solution of CUSO4. What is the error of the 
ammeter reading? Ans. The ammeter reads 0.46 ampere too 
low. 

167. Find the current density required to deposit a layer of 
copper o.oi inch in thickness on a flat cathode plate during 
5 hours from a solution of cupric sulphate. Ans. 34.4 amperes 
per square foot. 

Note. The specific gravity of copper is about 8.9. 

168. Find the time required for 10 amperes of current to 
liberate 2 cubic feet (5.074 grams) of hydrogen and one cubic 
foot of oxygen in an electrolytic cell like that shown in Fig. 157. 
Ans. 13.62 hours. 

Note. Consider that 0.000328 gram of copper is deposited per second per 
ampere from a solution of cupric sulphate (CUSO4), and consider one ampere 
liberates the same amount of sulphuric acid radical (SO4) from a solution of sul- 
phuric acid or of any sulphate. 

169. A voltaic cell which is free from local action gives a current 
of 1.5 amperes for 50 hours. Calculate the number of grams of 
zinc consumed. Ans. 91.5 grams. 

Note. The zinc consumed in a voltaic cell by voltaic action is equal to the 
amount which would be deposited in an electrolytic cell by the current which the 
voltaic cell delivers. 

170. A chromic acid cell Is connected to the electrodes of an 
electrolytic cell like that shown in Fig. iii, and 125 grams of 



564 APPENDIX A. 

zinc is consumed during the time that 25 grams of copper is 
deposited on the electrode C from a solution of CUSO4. What 
portion of the zinc is consumed by local action? Ans. 79.4 
per cent. 

171. A gravity cell is used to give a steady current of o.i 
ampere continubuvsly, night and day, for 30 days. During this 
time 1,120 grams of copper sulphate crystals are used. Find: 
(a) The amount of copper sulphate crystals which is consumed 
by voltaic action, and {h) the amount of copper sulphate crystals 
which is consumed by local action. Ans. (a) 335 grams con- 
sumed by voltaic action and {h) 785 grams wasted by local action. 

Note. Copper sulphate crystals contain 5 molecules of water of crystallization, 
that is to say, the formula for copper sulphate crystals is CuS04 + 5H2O, so that 
249.6 grams of copper sulphate crystals contain 63.6 grams of copper. 

172. How much caustic soda (NaOH) is used up in a copper-oxide cell (see Art. 
158) while the cell is delivering 5 amperes for 24 hours, assuming local action to 
be non-existent? Ans. 179 grams. 

173. What is the reduction in weight of the copper oxide cathode in a copper- 
oxide cell after the delivery of 5 amperes for 24 hours? Ans. 35.8 grams. 

174. A lead storage cell delivers 10 amperes for 8 hours. Find the increase of 
weight of each electrode. Ans. The positive electrode gains 0.2105 pound, and 
the negative electrode gains 0.3156 pound. 

Note. One pound equals 453.6 grams. 

PROBLEMS. CHAPTER XIII. 

THE HEATING EFFECT OF THE ELECTRIC CURRENT. 

OHM'S LAW. 

175. A current of 0.5 ampere flowing through a glow lamp 
generates 150 calories of heat in 10 seconds. What is the 
resistance of the lamp? Ans. 252 ohms. 

176. The field coil of a dynamo contains 11,340 grams of 
copper (specific heat 0.094), weight of cotton insulation negligible. 
The resistance of the coil is 100 ohms. At what rate does the 
temperature of the coil begin to rise when a current of 0.5 ampere 
is started in the coil? Ans. 0.0056 centigrade degree per second. 

177. A given piece of copper wire has a resistance of 5 ohms, 
another piece of copper wire is 1.5 times as long but it has the 



PROBLEMS. 565 

same weight (and volume) as the first piece. What is its re- 
sistance? Ans. 11.25 ohms. 

178. A given spool wound full of copper wire 60 mils in diam- 
eter has a resistance of 3.2 ohms. An exactly similar spool is 
wound full of copper wire 120 mils in diameter; what is its 
resistance? Ans. 0.2 ohm. 

Note. The spool will contain half as many layers and half as many turns in 
each layer of the larger wire, and the mean length of one turn of wire is the same 
in each case. 

179. What is the resistance at 20° C. of 2 miles of commercial 
copper wire 300 mils in diameter? Ans. 1.22 ohm. 

180. Find the resistance at 20° C. of a copper conductor 100 
feet long having a rectangular section 0.5 inch by 0.25 inch. 

Ans. 0.00653 ohm. 

IT f d \^ 

Note. The area of a circle d mils in diameter is d^ circular mils, or — ( • I 

4 \ 1000/ 

square inches. Therefore the sectional area of a rectangular bar in square inches 

4,000,000 
must be multiplied by • to reduce to circular mils. 

IT 

181. What is the resistance at 20° C. of a wrought iron pipe 
20 feet long having one inch inside diameter and 1.25 inches 
outside diameter. Ans. 0.00206 ohm. 

Note. Use resistivity of pure annealed iron. 

182. Calculate the resistance in ohms of an arc lamp carbon 
0.5 inch in diameter and 12 inches long. Ans. 0.1207 ohm. 

183. Calculate the resistance of a column of 5 per cent solution 
of sulphuric acid at 18° C, the length of the column being 20 
centimeters and the sectional area being 12 square centimeters. 
Ans. 8 ohms. 

184. A coil of copper wire has a resistance of 5 ohms at 20° C, 
what is its resistance at 0° C, and what is its resistance at 90° C? 
Ans. 4.63 ohms; 6.297 ohms. 

Note. These results are calculated on the assumption that the curve h in 
Fig. 170 is a straight line whose inclination is 0.004 (see column ^ in the table on 
page 235). Now as a matter of fact the curve b in Fig. 170 is not a straight line, 
and the value 0.004 which is given in the table varies considerably for different 
samples of commercially pure copper. 



566 APPENDIX A. 

According to equation (72) of Art. 168 the resistance Rt of a piece of wire at 
t° C. should be calculated from the resistance at 20° C. (namely R20) by using the 
two equations: 

JR20 = Roil + 20/3) 
and 

Rt = Roil + tfi) 
from which we get: 



Rt = Ri 



( 1 +0 \ 

\I +20)3/ 



185. A wire has a resistance of 164.8 ohms at 0° C. and a resist- 
ance of 215.2 ohms at 85° C. What is the mean temperature 
coefficient between 0° C. and 85° C? Ans. 0.0036. 

186. The field coil of a dynamo has a resistance of 42.6 ohms 
after the dynamo has stood for a long time in a room at 20° C. 
After running for several hours the resistance of the coil is 51.6 
ohms. What is its temperature? Ans. y'j° C. 

187. The curves in Fig. 170 can be expressed more accurately by an equation 
of the form Rt = Roil -\- at + W^) than by the simpler equation Rt = i?o(i + ^t). 
A sample of very pure annealed platinum wire has a resistance of 124.3 ohms at 
0° C, 242.38 ohms at 250° C, and 338.8 ohms at 500° C, Find the values of the 
coefficients a and h. Ans. a = -\- 0.003945; h = — 0.000000584. 

188. A coil of very pure annealed platinum wire has a resistance of 24.62 ohms 
at 0° C, and when the wire is placed in a furnace and protected from the furnace 
gases by a porcelain tube it has a resistance of 120.8 ohms. Find the temperature 
of the furnace using the equation Rt = Roii + c/ + ht"^). Ans. t = 1205° C. 

Note. The curves in Fig. 170 cannot be thought of as approximately straight 
lines for very great temperature differences. 

189. A carbon-filament glow lamp has a resistance of 277 ohms 
at 0° C, and a resistance of 220 ohms at 1,000° C. What is the 
mean temperature coefficient of resistance of the filament be- 
tween 0° C. and 1,000° C? Ans. — 0.000206. 

190. A glow lamp takes 0.6 ampere when the electromotive 
force between its terminals is no volts. Find the power de- 
livered to the lamp and express it in horse-power. Ans. 0.0884 
horse-power. 

191. A motor takes 79.78 amperes of current from no-volt 
mains, and the motor-belt delivers 10 horse-power. What is the 
efficiency of the motor? Ans. 85 per cent. 

Note. Power output of motor in watts divided by power intake of motor in 
watts gives the efficiency of the motor. 



PROBLEMS. 567 

192. A so-called "25-watt, iio-volt" tungsten lamp takes 25 
watts of power when it is connected to 110- volt supply mains. 
How much current does the lamp take, and what is the resistance 
of the lamp filament while the lamp is burning? Ans. (a) 0.227 
ampere; (b) 473.6 ohms. 

193. A motor to deliver 10 horse-power has an efficiency of 
89 per cent. The motor is supplied with current at no volts 
across its terminals. Find the full-load current of the motor 
and find the size of rubber insulated wire required (according to 
Insurance Rules) to deliver current to the motor. Ans. 76.2 
amperes; number 3 wire B. & S. gauge. 

Note. See table on page 230. 

194. When electrical energy costs 11 cents per kilowatt-hour 
how much does it cost to operate for 10 hours a lamp which 
takes 0.227 ampere from no-volt supply mains? Ans. 2.75 
cents. 

195. Find the cost of energy for operating a 5 horse-power 
motor at full load for 10 hours, the efficiency of the motor being 
85 per cent and the cost of energy being 6 cents per kilowatt- 
hour. Ans. $2.63. 

196. Assume the actual cost of electrical energy delivered to a 
street car to be 0.8 cent per kilowatt-hour. Find the cost of 
developing 100,000 British thermal units for heating the car 
first by an electrical heater in which all of the heat generated is 
available for heating, and second by burning coal costing $6 per 
ton (2,000 pounds) and giving 14,000 British thermal units per 
pound of which 30 per cent, say, is lost by incomplete combustion 
and by flue gas losses. Ans. (a) $2.35; (b) $1.43. 

197. One cubic foot of good illuminating gas costing one tenth 
of a cent gives about 600 British thermal units when it is burned, 
and about 20 per cent of the heat of a burner is taken up by the 
water in a tea kettle. On the other hand about 70 per cent of 
the heat given off by an ordinary electrical heater is given to a 
tea kettle which completely covers the hot disk of the heater, 
and electrical energy for domestic use costs, say, 10 cents per 



568 APPENDIX A. 

kilowatt-hour. What is the cost of bringing 2 gallons of water 
from 15° C. to 100° C. by a gas burner and what is the cost by 
electric heater? Ans. (a) 2.31 cents; (b) 10.74 cents. 

198. When a certain dynamo electric generator is delivering 
no current it takes 1.75 horse-power to drive it. When the 
generator delivers 150 amperes it takes 25 horse-power to drive it. 
Calculate the electromotive force of the generator on the assump- 
tion that all of the additional power required to drive it is used 
to maintain the current of 150 amperes. Ans. 115.7 volts. 

199. A coil of wire of which the resistance is to be determined 
is connected to iio-volt direct-current supply mains in series 
with an ammeter and a suitable rheostat, and a voltmeter is 
connected across the terminals of the coil. The ammeter reads 
13 amperes and the voltmeter reads 80.6 volts. What is the 
resistance of the coil? Ans. 6.2 ohms. 

200. A gravity Daniell cell of which the electromotive force Is 
1.07 volts and the resistance is 2.1 ohms is connected to a wire 
circuit of which the resistance is 5 ohms, (a) What current is 
produced? (b) What is the electromotive force between the 
terminals of the cell? (c) What is the electromotive force drop 
in the cell? Ans. (a) 0.15 ampere; (b) 0.75 volt; (c) 0.32 volt. 

201. A voltmeter connected across the terminals of a set of 60 
storage battery cells connected in series reads 120.4 volts when 
the battery is delivering no current, and the voltmeter reading 
falls instantly to 112.25 volts when the battery begins to deliver 
15 amperes of current. What is the resistance of the battery? 
Ans. 0.55 ohm. 

Note. When a battery continues to deliver current the voltage falls off because 
of polarization. The sudden drop of voltage at the instant that current delivery 
begins is due almost entirely to the battery resistance. 

202. A voltmeter connected across the terminals of a battery 
reads 15 volts when the battery is not delivering current (except 
the negligible current which flows through the voltmeter), and 
the voltmeter reading drops suddenly to 9 volts when a wire 
circuit having a resistance of 6 ohms is connected to the battery. 
What is the resistance of the battery? Ans. 4 ohms. 



PROBLEMS. 569 

203. A dynamo electric generator having an electromotive force 
of 115 volts between its terminals delivers 200 amperes to a group 
of glow lamps 1,000 feet distant from the generator. Find the 
diameter in mils of the copper wire required in order that 95 
per cent of the power output of the generator may be delivered 
to the lamps. Ans. 890 mils. 

Note. If 95 per cent of the power output of the generator is delivered to the 
lamps then the electromotive force between the mains at the lamps must be 95 
per cent of 115 volts, or 109.25 volts, so that the voltage drop over the line is 5.75 
volts. 

204. What size of copper wire is required to deliver current at 
no volts to a 10 horse-power motor of 85 per cent efficiency; 
the motor being 2,000 feet from the generator, and the electro- 
motive force across the generator terminals being 125 volts? 
Ans. 470 mils in diameter. 

Note. Find the power in watts delivered to the motor, then find the current, 
and then proceed to find the resistance oi the line from a consideration of the line 
drop, and so on. 

205. A motor is to receive 100 kilowatts of power from a gen- 
erator at a distance of 15 miles. A loss of 10 per cent of generator 
voltage (or 10 per cent of the generator output of power) is to 
be permitted in the transmission line. Find the generator 
voltage which must be provided for in order that copper trans- 
mission wires 200 mils in diameter may be used. Ans. 6,763 volts. 

206. If a 10 per cent line loss is allowed as in the previous prob- 
lem, but if the generator voltage is doubled (making it 13,526 
volts), what size of copper transmission wires would be used to 
deliver 100 kilowatts at a distance of 15 miles? Ans. 100 mils 
in diameter. 

Note. It is worthy of note that by doubling the generator voltage the cost of 
the copper required to transmit a given amount of power over a given distance is 
quartered. Compare this problem and the preceding one with the next one. 

207. What is the weight of 30 miles of copper wire 200 mils in 
diameter and what will it cost at 15 cents per pound? What is 
the weight of 30 miles of copper wire 100 mils in diameter and 
what will it cost at 15 cents per pound? Ans. (a) 24,420 pounds, 
$3,663.00; {h) 6,105 pounds, $9i5-75- 



570 APPENDIX A. 

208. A millivoltmeter has a resistance of 15.4 ohms. What 
resistance must be connected in series with the instrument so 
that the scale reading may give volts instead of millivolts? 
Ans. 15,384.6 ohms. 

209. Three lamps (or other units) are connected in series to 
1 10- volt mains, the resistances of the lamps are 10 ohms, 8 ohms 
and 4 ohms respectively, find the voltage across the terminals 
of each lamp. Ans. 50 volts, 40 volts and 20 volts. 

210. Three resistances of 4, 4 and 2 ohms respectively are 
connected in parallel ; and two resistances of 6 ohms and 3 ohms 
respectively are connected in parallel. The first combination 
is connected in series with the second combination, and to a 
battery of negligible resistance and of which the electromotive 
force is 3 volts. What is the current in the 2 ohm resistance and 
what is the current in the 3 ohm resistance? Ans. 0.5 ampere 
and 0.66 ampere respectively. 

211. An ammeter has a resistance of 0.05 ohm. The instru- 
ment is provided with a shunt so that the total current through 
instrument and shunt is 10 times the current through the am- 
meter itself. What is the resistance of the shunt? Ans. 0.00556 
ohm. 

212. The scale of a direct-reading millivoltmeter has 100 
divisions, each division corresponding to one thousandth of a 
volt between the terminals of the instrument. The instrument 
is connected to the terminals of a low-resistance shunt, and 
each division on the instrument scale corresponds to 0.25 ampere 
in the shunt. What is the resistance of the shunt? Ans. 0.004 
ohm. 

213. A voltmeter which has a resistance of 16,000 ohms is 
connected in series with an unknown resistance R to iio-volt 
supply mains, and the reading of the voltmeter is 4.3 volts. 
What is the value of R? Ans. 393,300 ohms. 

214. A 40-mile telegraph line is disconnected from ground at 
both ends, the line is then connected to ground at one end 
through a 220-volt battery and a direct-reading voltmeter of 



PROBLEMS. 571 

which the resistance is 16,000 ohms, and the voltmeter reads 
2.9 volts. What is the insulation resistance of the 40-mile 
telegraph line, and what is the insulation resistance of one mile 
of the line? Ans. 1,197,000 ohms; 47,880,000 ohms. 

Note. The voltmeter is here used as an ammeter, the current through the 
instrument is equal to its reading in volts divided by its resistance in ohms. The 
current thus measured flows through the voltmeter and leaks from the line to the 
ground through the very high resistance of the insulators. The entire resistance 
of the circuit may be found by using Ohm's law inasmuch as the voltage of the 
battery is given. The resistance of the battery and of the wires is negligible. 

Consider the portion of the circuit through which the measured current flows 
from telegraph wire to ground. The sectional area of this portion is proportional 
to the length of the telegraph line, and therefore the sectional area corresponding 
to one mile of line is one fortieth of the sectional area corresponding to 40 miles 
of line. 

PROBLEMS. CHAPTER XIV. 

INDUCED ELECTROMOTIVE FORCE. 

215. The armature of a direct-current motor has a resistance of 
0.064 ohm, and when the motor is running under full load a 
current of 81 amperes is forced through the armature from 11 o- 
volt supply mains. What is the back electromotive force in the 
armature? Ans. 104.8 volts. 

216. How much power is delivered to the motor armature in 
the previous problem? How much power is spent in heating the 
armature wires? How much power is used in forcing the current 
through the armature in opposition to the back electromotive 
force? 

217. A two-pole direct-current dynamo has 560 wires on out- 
side of its armature (= Z), it runs at a speed of 900 revolutions 
per minute, and it gives an electromotive force of 125 volts (at 
zero current output). What amount of magnetic flux passes 
from north pole face into armature core and from armature core 
into south pole face? 

218. The intensity of the magnetic field in the gap spaces 
between field pole-faces and armature core of a direct-current 
dynamo is 6,000 gausses and the length of iron parts of armature 



572 APPENDIX A. 

core and pole- face is 15 centimeters. The diameter of the arma- 
ture (measured out to centers of armature wires) is 30 centi- 
meters, and the speed of the armature is 900 revolutions per 
minute. What is the velocity of the armature wires as they cut 
the lines of force of the magnetic field in the gap spaces and 
what amount of electromotive force is induced in each wire? 
. 219. One of the magnets of problem 143 is placed through a 
short coil of wire containing 20,000 turns of wire, and jerked 
quickly out of the coil so that the magnetic flux through the 
coil drops to zero in o.oi second. What is the average value of 
the electromotive force which during the 0.0 1 second is induced 
in the coil of wire? 

Note. Find the amount of magnetic flux through the coil (see problem 150), 
divide this by the time during which it is reduced to zero to get the rate of change of 
flux. See latter part of Art. 186 where it is shown that the induced electromotive 
force in abvolts in each turn of wire of a coil is equal to the rate of change of 
magnetic flux through the coil. 

220. The winding of an electromagnet has a resistance of 22 
ohms, and when the winding is connected across no- volt supply 
mains the current in the coil rises from zero at the beginning to 
5 amperes ultimate value. The current must therefore pass in 
succession the values i ampere, 2 amperes, 3 amperes and 4 
amperes. Consider the instant when the current is passing the 
value of 2 amperes, and calculate the values of the following 
quantities at this instant: (a) The rate at which work is being 
delivered to the coil, {h) The rate at which work is being spent to 
generate heat in the coil, (c) The rate at which work is being 
spent in overcoming a back electromotive force in the coil due to 
the increasing magnetism of the rod, and {d) The value of the 
back electromotive force due to the increasing magnetism of the 
rod. Ans. (a) 220 watts; {h) 88 watts; (c) 132 watts; (d) 66 
volts. 

Note. In this arrangement work is spent only in generating heat in the coil 
in accordance with Joule's law and in magnetizing the rod. 

The portion of the electromotive force of 220 volts which is being used to over- 
come resistance is equal to RI, where R is the resistance of the coil and / is the 



PROBLEMS. 573 

value of the current at any instant. The remainder of the electromotive force of 
220 volts overcomes the back electromotive force and is equal thereto. 

221. The coil in the previous problem has io,ooo turns of 
wire. Find how fast the magnetic flux through the coil is in- 
creasing as the current is passing the value of 2 amperes. Ans. 
660,000 maxwells per second. 

222. A boat is started from rest by a steady force of no 
poundals, and after some time the boat reaches a steady speed of 
5 feet per second. The velocity must therefore pass in succession 
the values i foot per second, 2 feet per second and so on. Con- 
sider the instant when the velocity is passing through the value 
of 2 feet per second, and assume that the part of the total force 
which is being used to overcome friction is proportional to the 
velocity of the boat; and calculate the following quantities at 
this instant : (a) The rate at which work is delivered to the boat, 

(b) The rate at which work is being spent in overcoming friction, 

(c) The rate at which work is being spent in increasing the velocity 
of the boat, and (d) The value of the inertia reaction of the boat. 
Ans. 220 foot-poundals per second; (b) 88 foot-poundals per 
second; (c) 132 foot-poundals per second; (d) 66 poundals. 

Note, no poundals overcomes friction at 5 feet per second, and, according to 
our assumption, proportionately less force is used to overcome friction at 2 feet 
per second. The inertia reaction of an accelerated body is equal and opposite to 
the force which is producing acceleration. 

223. The electromotive force between the terminals of a shunt 
generator is 120 volts, and the resistance of the shunt field 
winding is 22 ohms. How much current flows through the shunt 
field winding? If the generator delivers 80 amperes to a group 
of lamps, how much current is delivered by the armature? 
Ans. (a) 5.45 amperes; (b) 85.45 amperes. 

Note. See note to problem 225. 

224. The resistance of the field winding of a series dynamo is 
0.08 ohm, the dynamo, operating as a generator, delivers 80 
amperes, and the electromotive force between the brushes is 
125 volts. What is the electromotive force between the terminals 



574 APPENDIX A. 

of the machine, that is, between the points of attachment of the 
external circuit? Ans. 118.6 volts. 

Note. The series dynamo is seldom used as a generator in practice. The series 
dynamo is frequently used as a motor, in street cars, for example. 

225. Find the power expended in field excitation in the two 
cases specified in problems 223 and 224. Ans. (a) 654 watts; 
{h) 512 watts. 

Note. After the current in the field winding and the "strength" of the field 
magnet become steady in value, all of the power delivered to the field winding 
reappears as heat in the winding in accordance with Joule's law. Therefore Ohm's 
law applies to the field winding. No power would be required to maintain the 
magnetism of the field magnet if a field winding of zero resistance could be obtained. 
When, however, the field magnet is being magnetized (during a small fraction of a 
second at the beginning) then some of the power delivered to the field winding does 
not reappear as heat in accordance with Joule's law, but is used to establish the mag- 
netism. Thus if E is the voltage across the terminals of a magnet winding, I the 
current in the winding, and R the resistance of the winding, then EI is the rate 
at which work is delivered to the winding and RI"^ is the rate at which heat is 
developed in the winding. Now the ultimate value of / is EjR, and when this 
ultimate value of / is reached we have EI = RP, but before I reaches this 
ultimate value EI is greater than RD and the excess is the power which at each 
instant is being used to establish the magnetism. 

226. The large water-wheel-driven alternators in the upper 
power house at Niagara Falls have a speed of 250 revolutions per 
minute and they give an alternating electromotive force having a 
frequency of 25 cycles per second. How many field poles are 
there in the field magnet of one of these machines? 

227. The first of the large steam-driven power plants in New 
York City were equipped with alternators which were mounted 
directly on the crank shafts of large reciprocating engines 
running at a speed of 75 revolutions per minute. The frequency 
of the alternating electromotive force which is generated is 25 
cycles per second. How many field poles are there in one of the 
field magnets of one of these alternators. 

228. A fixed percentage of the power output, Ely of a gen- 
erator is to be lost as RP loss in a transmission line. Show, on 
the basis of this assumption, that the amount of copper wire in 
pounds required to transmit a given amount of power over a 



PROBLEMS. 575 

given distance is quartered if the generator voltage E is doubled. 

Note. If E is doubled, then / will be halved inasmuch as the power output 
EI is a given amount. Also the line loss in watts is fixed in value, but the line 
loss is RD where R is the resistance of the line (including both wires). There- 
fore, since I is halved and Rl"^ is unchanged, it is evident that R is quadrupled. 
From this point the argument can be carried forward on the basis of equation (71) 
of Art. 167. 

229. A shunt generator is driven at a speed of i ,200 revolutions 
per minute, and it gives an electromotive force of no volts be- 
tween its brushes (armature current negligibly small) with a total 
of 56 ohms in its shunt field circuit. If the generator is driven at 
a speed of 1,500 revolutions per minute how much additional 
resistance will have to be connected in the shunt field circuit in 
order that the electromotive force may be increased in proportion 
to the increase of speed so as to be 137.5 volts? Ans. 14 ohms. 

Note. In order that the electromotive force may be increased in proportion 
to the increase of speed, the value of $ must remain unchanged and therefore the 
current in the shunt field winding must remain unchanged. 

230. The electromotive force of a shunt generator (armature 
current negligibly small) decreases from no volts to 93 volts 
when the speed of the generator is reduced from i ,000 revolutions 
per minute to 900 revolutions per minute. The armature flux $ 
at the higher speed is 1,000,000 lines, {a) What would the elec- 
tromotive force of the generator be at the lower speed if the 
armature flux were unchanged? (&) What is the value of the 
armature flux at the lower speed? Ans. (a) 99 volts; (h) 939,390 
lines. 

231. The resistance of the armature of a direct-current gener- 
ator (including brushes and brush contacts) is 0.14 ohm. The 
electromotive force induced in the armature {^Zn) is 120 volts 
and the armature current is 70 amperes. Find the value of the 
electromotive force between the brushes. Ans. no. 2 volts. 

Note. See Art. 178. 

232. A coil of wire has a resistance of 20 ohms and an induc- 
tance of 0.6 henry. When connected to no-volt supply mains 
the current rises to a steady value of 5.5 amperes. How fast does 



576 APPENDIX A. 

the current start to increase in the coil when it is first connected 
to the 1 10- volt supply mains and how long would it take for the 
current to reach its steady value of 5.5 amperes if it continued to 
increase at this initial rate? Ans. 183.3 amperes per second; 
0.0289 second. 

Note. When the current is zero no portion of the supply voltage is used to over- 
come resistance, it is all used to make the current increase. The time which would 
be required for the current to reach its final steady value at its initial rate of 
increase is called the time constant of the circuit. 

233. From the definition of the time constant of a circuit as 
given in the note to the previous problem show that the time 
constant of any circuit is equal to LjR where L is the inductance 
of the circuit and R is the resistance of the circuit. 

Note. Find an expression for the initial rate of increase of the current in the 
circuit for any value E of supply voltage, and then find how long it would take 
at this rate to reach the value EjR. 

234. Find the rate of growth of current in the coil of problem 
232 at the instant that the current is passing the value of 3 
amperes. Ans. 83^ amperes per second. 

235. A current is left to die away in a circuit of which the 
resistance is 0.6 ohm and the inductance is 0.05 henry. Find 
the rate of decrease of the current as it passes the value of 10 
amperes. Ans. 120 amperes per second. 

Note. In this problem no electromotive force is supposed to act on the circuit; 
the resistance drag, however, is RI volts and it is this drag which causes the 
current to decrease. 

PROBLEMS. CHAPTER XVI. 

ELECTRIC CHARGE AND THE CONDENSER. 

236. During 0.03 second a charge of 15 coulombs passes 
through a circuit. What is the average value of the current 
during this time? Ans. 500 amperes. 

237. Suppose the strength of a current in a circuit to increase at 
a uniform rate from zero to 50 amperes in 3 seconds. Find the 
number of coulombs of charge carried through the circuit by the 
current during the 3 seconds. Ans. 75 coulombs. 



PROBLEMS. , 577 

Note. The average value of the current during the 3 seconds is half the sum of 
the initial and final values, because the current changes at a constant rate. 

238. A condenser of which the capacity is known to be 5 micro- 
farads, is charged by a Clark standard cell of which the electro- 
motive force is 1.434 volts, and then discharged through a 
ballistic galvanometer. The throw of the ballistic galvanometer 
is observed to be 15.3 scale divisions. What is the reduction 
factor of the galvanometer? Ans. 469 X io~^ coulombs per 
division. 

239. A condenser of unknown capacity is charged by 10 Clark 
cells in series, giving an electromotive force of 14.34 volts, 
and then discharged through the ballistic galvanometer specified 
in the previous problem. The throw of the ballistic galvanometer 
is observed to be 18.6 divisions. What is the capacity of the 
condenser? Ans. 0.608 microfarad. 

240. An electromotive force acting on a condenser increases at 
a uniform rate from zero to 100 volts during an interval of 0.005 
second. The capacity of the condenser is 20 microfarads. Find 
the value of the current. Ans. 0.4 ampere. 

241. Two parallel metal plates at a fixed distance apart with 
air between are charged as a condenser, and discharged through 
a ballistic galvanometer. The plates are then submerged in 
turpentine and again charged and discharged through the same 
ballistic galvanometer. The charging electromotive force is 
the same in each case, and the throw of the ballistic galvano- 
meter is observed to be 7.6 divisions in the first instance and 16.7 
divisions in the second instance. Find the inductivity of the 
turpentine. 

242. A condenser consists of two square flat metal plates each 
2 meters by 2 meters, and the plates are i centimeter apart. 
What is the capacity of the condenser in farads? Ans. 0.003536 
microfarad. 

243. What would be the capacity of the condenser specified 
in the previous problem if the whole were submerged in light 
petroleum (kerosene)? Ans. 0.0072 microfarad. 

38 



578 APPENDIX A. 

244. A condenser Is to be built up of sheets of tin foil 12 
centimeters by 15 centimeters. The overlapping portions of the 
sheets are to be 12 centimeters by 12 centimeters. The sheets 
are to be separated by leaves of mica 0.05 centimeter thick. 
How many mica leaves and how many tin foil sheets are required 
for a one-microfarad condenser? Assume the inductivity of 
the mica to be equal to 6. Ans. Mica leaves, 655; tin foil 
sheets, 656. 

245. A condenser is made of two flat metal plates separated by 
air. Its capacity is 0.003 niicrofarad. Another condenser has 
plates twice as wide and twice as long. These plates are sepa- 
rated by a plate of glass (inductivity 5) which is four times as 
thick as the air space in the first condenser. What is the capacity 
of the second condenser? Ans. 0.015 microfarad. 

246. The two metal plates arranged as in problem 242 are 
charged by connecting them to iio-volt direct-current supply 
mains. Find the amount of charge on each plate, and find the 
energy of the charged condenser. Ans. 0.389 microcoulomb ; 
21.4 microjoules or 214 ergs. 

247. The charged plates which are specified in the previous 
problem are insulated so that the charge on each plate must 
remain constant, and one of the plates is then moved until the 
plates are 2 centimeters apart. What is the voltage between the 
plates after the movement? What is the energy of the charged 
condenser after the movement? Ans. 220 volts; 428 ergs. 

248. The increase of energy from 214 ergs to 428 ergs due to 
the movement of the plate as specified in problem 247, is equal to 
the work done in pulling the plates apart against their mutual 
force of attraction F. Find the value of F. Ans. 214 dynes. 

Note. Consider that the work done against the force of F dynes is equal to F 
multiplied by the movement in centimeters. 

The force of attraction F is independent of the distance apart of the plates 
for the given amount of charge q, provided the distance between the plates is small 
in comparison with the size of the plates. 

The following problems can be solved by the method illustrated in this problem 
and the preceding one. 

249. Find the force of attraction of the plates referred to in prob- 



PROBLEMS. 579 

lem 248 when the charge on each plate is such as would be pro- 
duced by connecting the plates to iio-volt supply mains when 
the plates are 2 centimeters apart. Ans. 53.5 dynes. 

250. Find the force of attraction of the plates referred to 
in problem 248 when the plates are connected to 1,1 00- volt supply 
mains when the distance between the plates is i centimeter. 
Ans. 21,400 dynes. 

251. The two plates referred to in problem 248 are submerged 
in kerosene (inductivity 2.04). Find the force with which they 
attract each other when they are at a distance of i centimeter 
apart and connected to 1,100- volt supply mains. Ans. 43,660 
dynes. 

PROBLEMS. CHAPTER XVII. 

LIGHT AND SOUND DEFINED. 

252. Take a pair of dividers and space the points until one 
can just fail to distinguish two points (a) When the points are 
touched to the end of the finger, {h) When the points are touched 
to the back of the hand, and {c) When the two points are touched 
to the back of the neck. What is the distance between the points 
in each case? 

Note. A single nerve fiber of a nerve of touch sensitizes a certain sized spot on 
the skin, and the distances (a), (&) and (c) are the approximate diameters of these 
unit spots at the respective places on the body. 

253. Place two minute drops of mercury close together in full 
sunlight on a dark block of wood, and walk away from the block 
until the two bright specks barely become indistinguishable as a 
pair. Measure the distance d between the mercury drops, and 
your distance D therefrom, and calculate the approximate 
diameter of the "unit spots" on the retina. 

Note. Let E be the diameter of the eye, front to back; the:.i the result required 
is approximately equal to <i X £/D. 

254. Ten seconds elapse between a flash of lightning and the 
first peal of thunder produced thereby. What is the distance 
from the observer to the nearest part of the lightning flash? 



580 



APPENDIX A. 



PROBLEMS. CHAPTER XVIII. 

WAVE MOTION. 

255. Find the wave-length of the wave train produced in 
dry air by a body making 256 complete vibrations per second. 

256. Twelve drops of water per second fall from a nozzle into 
a pool of water. The wave train produced on the surface of the 
pool travels at a velocity, say, of four feet per second. What is 
the wave-length of the train? 

257. Yellow light has a wave-length of approximately 59 
millionths of a centimeter. What is the frequency of a wave 
train of yellow light? 

Note. By frequency is meant the number of complete waves passing a given 
point per second. 

PROBLEMS. CHAPTER XIX. 

REFLECTION AND REFRACTION. 

258. A 2-inch cube of glass CC, Fig. 474, is placed as shown 

\S 



sight line 

>- 




Fig. 474. 

in front of a scale SS so that the scale can be seen through the 
cube at a, and by looking over the cube at h ; and the distance 
ah, as read off the scale, is 0.39 inch. Find the index of refraction 
of the glass of which the cube is made. Ans. 1.5. 

259. A block SS, Fig. 475a, of a substance of which the index 
of refraction is to be determined has a polished face and it is 



PROBLEMS. 



581 



submerged In carbon bisulphide LL In a glass vessel VV which 
has a flat front FF. A sheet of paper pp partly surrounds the 
vessel so as to give a uniform Illumination. Under these condi- 
tions a sharply defined line Is seen In the field of the telescope 
when the crystal Is turned so that the critical angle of total reflec- 
tion / Is as shown. The block 56* Is supported at the end of a 
vertical axis which projects above the surface of the carbon bisul- 
phide, passes through the center of a horizontal divided circle 
and carries a pointer which plays over a divided circle so that 
the angle turned to bring the block 55 from the position shown 
in Fig. 475a to the position shown In Fig. 4756 can be observed. 





This angle is thus found to be 125° and the Index of refraction 
of the carbon bisulphide Is 1.63. What is the Index of refraction 
of the substance 55? Ans. 1.445. 

Note. The arrangement shown in Fig. 475 is extensively used for quick and 
easy measurements of indices of refraction. It Is called a refractometer. A slightly 
modified form of the instrument can be used for measuring the index of refraction 
of a drop of liquid clinging to a glass surface. 

260. The line WW, Fig. 476, is the position which would be 
reached at a given instant by the wave CD were it not for the 
passage from glass Into air. Make a drawing showing Huygens' 



582 APPENDIX A. 

construction for the refracted wave at the given instant, index 
of refraction of the glass being 1.5 and angle B being 60°. 




Fig. 476. 

PROBLEMS. CHAPTER XX. 

SIMPLE LENSES. 

261. An object, represented by a vertical arrow 10 centimeters 
long, is at a distance of 50 centimeters from a converging lens of 
which the focal length is 15 centimeters. Make carefully a 
drawing, to scale, showing the position and size of the image. 
Is the image real or virtual? 

Note. See Art. 269. 

262. The object in problem 261 is placed 10 centimeters from 
the lens. Make carefully a drawing, to scale, showing the posi- 
tion and size of the image. Is the image real or virtual? 

263. An object, represented by a vertical arrow 10 centimeters 
long, Is at a distance of 50 centimeters from a diverging lens of 
which the focal length is 15 centimeters. Make carefully a 
drawing, to scale, showing the position and size of the image. 
Is the image real or virtual? 

264. Rays of light coming from the left converge towards an 
image a but never reach the image because they pass through 
a converging lens of which the focal length is 15 centimeters. 
The image a is 50 centimeters to the right of the lens and it is 



PROBLEMS. 583 

represented by a vertical arrow 10 centimeters long. Make 
carefully a drawing, to scale, showing the position and size of the 
image of a. Is this image real or virtual? 

265. Rays of light coming from the left converge towards an 
image a but never reach the image because they pass through 
a diverging lens of which the focal length is 15 centimeters. The 
image a is 50 centimeters to the right of the lens and it is repre- 
sented by a vertical arrow 10 centimeters long. Make carefully 
a drawing, to scale, showing the position and size of the image 
of a. Is this image real or virtual? 

266. The image a in the previous problem is 20 centimeters 
to the right of the lens. Make carefully a drawing, to scale, 
showing the position and size of the image of a. Is this image 
real or virtual ? 

267. Two simple converging lenses each having a focal length 
of 15 centimeters are placed co-axially at a distance of 10 centi- 
meters apart, and an object represented by a vertical arrow 10 
centimeters long is placed at a distance of 25 centimeters from 
the middle point of the two lenses. Make carefully a drawing, 
to scale, showing the position and size of the image produced by 
the joint action of both lenses. 

Note. Construct image produced by first lens, and then construct the image of 
this image produced by the second lens. 

268. A lens is made of glass of which the index of refraction is 
1.6. The diameter of the lens (measured to sharp edge) is 5 
centimeters and the thickness of the lens at the center is 0.5 
centimeter. What is its focal length? 

269. A lens which comes to a sharp edge has a diameter of 10 
centimeters and a thickness at center of 0.4 centimeter. The 
focal length of the lens Is 62.5 centimeters. What is the index 
of refraction of the glass? 

270. Two very thin simple lenses are placed very close together 
and used as a single lens. Show that the reciprocal of the focal 
length of the combination is equal to the sum of the reciprocals 
of the focal lengths of the respective simple lenses. 



584 



APPENDIX A. 



Note. This demonstration is perhaps most easily developed by considering 
that the combination retards the central part of a wave by the amount h -\- k, 
where h and k are the retardations of central part of wave by the respective lenses. 

The reciprocal of the focal length of a lens is called its "power," and this demon- 
stration shows that the "power" of two lenses placed close together is the sum of 
their individual "powers." See definition of diopter in Art. 274. 

271. An object 0, Fig. 477, is 10 meters in length. The 
focal length of lens A is 100 centimeters, and the focal length of 
lens ^ is 2 centimeters. Find the position and size of the image 



F 



axis 



_ 500 meters _ _ 



^ 101.8 cm. j 

Fig. 477. 



of formed by the combination. Ans. The image is real and 
erect, it is 8 centimeters to the right of B, and its length is 
10.02 centimeters. 

Note. This problem shows how an ordinary simple astronomical telescqpe may 
be used to project a large image of the sun on a screen. 

The dimensions involved in this arrangement make it very inconvenient to 
solve the problem by making a drawing. It is intended that this problem be 
solved by the use of equation (100) of Art. 272. 

272. An object 0, Fig. 478, is 10 meters in length. The 
focal length of lens A is 20 centimeters, and the focal length of 

A 



k 

axis 



'• 500 meters J, 18.2 Ctn, i 

Fig. 478. 

lens B {s2 centimeters (negative) . Find the position and size of 
the image of formed by the combination. Ans. The image is 



PROBLEMS. 585 

real and inverted, it is 18.83 centimeters to the right of B, and 
its length is 4.166 centimeters. 

Note. It is intended that this problem be solved by the use of equation (lOo) 
of Art. 272. The arrangement in Fig. 478 (an opera glass with the eye piece pulled 
out too far for looking) is exactly the same thing as the telephotographic lens. 
See Art. 296, section c, Fig. 373. 

373. A photographic lens of 15 centimeters focal length gives 
a sharp image of very distant objects when the slider carrying the 
lens is in a marked position. Calculate the distance the slider 
must be moved so that an object two meters in front of the lens 
may be in sharp focus on the photographic plate. Ans. 1.2 
centimeters forwards. 

Note. In solving this and the following problems treat the lens as if it were a 
simple lens. The lens must be 15 centimeters in front of the photographic plate 
for very distant objects, and h centimeters in front of the plate for an object at 
distance a in front of the lens. The distance h is greater than 15 centimeters, 
and the difference & — 15 is the required distance that the lens must be moved. 

374. A projection lantern takes a transparent slide or picture 
7X7 centimeters, and the focal length of the lantern object 
glass is 20 centimeters. It is desired to project on the screen an 
image of the slide two meters square. Required the distance 
from screen to lens. Ans. 592 centimeters. 

375. A projection lantern is to be used with its object glass 3 
meters from a screen, and it is desired to project a 7 X 7 centi- 
meter slide as a picture 2 meters square. What focal length 
object glass is required? Ans. 10.46 centimeters. 

376. A far sighted person sees distinctly an object at a distance 
of 200 centimeters or more. Find the power in diopters of a 
spectacle lens which will enable this person to see clearly an 
object at a distance of 25 centimeters or more from the eye. Ans, 
3.5 diopters positive. 

Note. The problem is to have a lens in front of the eye (distance from lens to 
eye being considered as negligibly small) so as to form a virtual image at 200 
centimeters of an object at 25 centimeters from the eye. It is intended that this 
and the following problem be solved by equation (100) of Art. 272. 

377. A near sighted person can see an object distinctly when 
it is not more than 15 centimeters from the eye. Find the power 



586 APPENDIX A. 

in diopters of a spectacle lens which will enable this person to 
see clearly an object which is not more than 200 centimeters from 
the eye. Ans. 6.17 diopters negative. 

Note. The problem is to have a lens in front of the eye which will form a virtual 
image at 15 centimeters from the eye of an object 200 centimeters from the eye. 

378. A pocket magnifier has 3 separate lenses of which the 
focal lengths are i, 2 and 4 centimeters, respectively. What is 
the magnifying power of each : (a) when the eye is accommodated 
for a distance of 25 centimeters, and {h) when the eye is accom- 
modated for parallel rays? Ans. {a) 26, 13.5 and 7.25 diameters; 
ih) 25, 12.5 and 6.25 diameters. 

379. An object o.i millimeter in diameter, actual size, is looked 
at through a microscope of which the magnifying power is 30 
diameters. What is the diameter of a drawing of the object 
which when placed at a distance of 3 meters from the naked eye 
will appear the same size as the object as seen through the mi- 
croscope? Ans. 3.6 centimeters. 

380. The three lenses of the magnifier which is described in 
problem 378 are used together as one lens. What is the magni- 
fying power of the combination with the eye accommodated for a 
distance of 25 centimeters, assuming the lenses to be indefinitely 
near together? Ans. 44.75 diameters. 

381. A compound microscope has an objective of which the 
focal length is 2 millimeters, an eye-piece of which the focal 
length is 10 millimeters, and the distance from center of objec- 
tive to the plane of the image {h in Fig. 328 of Art. 2'j'j) is 150 
millimeters. Calculate the magnifying power of the instrument 
on the assumption that the observer's eye is accommodated for 
a distance of 25 centimeters. Ans. 1,924 diameters. 

Note. The limit of effective magnifying power of a microscope is about 900 
diameters. With a magnifying power of 1924 diameters the image would be 
perceptibly blurred even if the microscope itself were perfect. See Art. 296, 
section d. 

382. The object glass of the great telescope of the Lick Obser- 
vatory is 1 ,500 centimeters in focal length. What is the magnify- 
ing power of the telescope when an eye piece having a focal length 



PROBLEMS. 587 

of 2 centimeters is used, the observer's eye being accommodated 
for parallel rays? 

383. The image formed by the object glass is at the focal point 
of the eye piece when the eye is accommodated for parallel rays 
as specified in problem 382. How far must the eye piece be 
drawn out from this position to project a sharp image of the 
sun on a screen three meters back of the eye piece? How large 
would the image of a spot on the sun be on this screen, the 
spot being, say, o.ooi of the sun's diameter (0.03 minute of arc)? 
Ans. 0.0134 centimeter, 1.96 centimeters size of image of spot on 
screen. 

384. An observer looks through the Lick telescope (using an eye piece of 2 
centimeters focal length) at a measuring stick 600 meters from the object glass, 
and with the other eye he looks directly at a scale of centimeters 30 centimeters 
from the eye. What length of this near-by scale will apparently be covered by one 
centimeter of the distant stick? Ans. 0.400 centimeter. 

Note. The one eye being accommodated for a distance of 30 centimeters, the 
other eye is unconsciously accommodated for 30 centimeters also. 

PROBLEMS. CHAPTER XXH. 

LENS IMPERFECTIONS. 

385. A lens has a free diameter of 2 centimeters and a focal 
length of 12 centimeters. What is its numerical aperture? The 
object glass of the great telescope of the Lick Observatory is 
92 centimeters free diameter and 1,500 centimeters focal length. 
What is its numerical aperture? 

386. A photographic lens of 15 centimeters focal length forms 
a satisfactory image over the whole of a photographic plate 
16 X 22 centimeters. What is the field angle of the lens? 
Ans. 84.4 degrees. 

387. The angular diameter of the sun as seen from the earth 
is half a degree. What is the diameter of the image of the sun 
formed by the object glavSs of the great telescope of the Lick 
Observatory? Ans. 13.05 centimeters. 

Note. Why is it that a long-focal-length lens of given diameter does not act 
satisfactorily as a "burning glass"? 



588 APPENDIX A. 

388. A photographic lens with a numerical aperture of J 
gives a good photograph with an exposure of, say, 0.02 second. 
What exposure would be required with a lens working with an 
//16 diaphragm or stop? Ans. 0.142 second. 

PROBLEMS. CHAPTER XXIV. 

INTERFERENCE AND DIFFRACTION. 

389. A glass plate 0.00004 centimeter thick is illuminated by a 
beam of parallel rays of light making an angle of 30° with the 
normal to the plate. The index of refraction of the glass is 1.5. 
What wave-lengths of the light will be strengthened by inter- 
ference within the region of the visible spectrum? Ans. 
X = 66 X io~^ centimeter and 39.6 X io~^ centimeter. 

390. The distance apart, center to center, of the slits in the 
grating AB, Fig. 400, is 0.0003 centimeter. The light TT is 
white light and the focal length of the lens LL is 50 centimeters. 
Calculate the distances from the point F to the two points 
F\ and from the point F to the two points F'\ and so on, for 
violet light (X = 40 X io~^ centimeter) and for red light (X 
= 75 X io~^ centimeter). Ans. Distances F to F\ for violet 
light 6.738 centimeters, for red light 12.91 centimeters; F to 
F^\ for violet light 13.83 centimeters, for red light 28.87 centi- 
meters. 

PROBLEMS. CHAPTER XXV. 

POLARIZATION AND DOUBLE REFRACTION. 

391. At what angle must a beam of rays strike a clean water 
surface in order that the reflected beam may be completely 
polarized? The index of refraction of water is 4/3. 

392. A solution of cane sugar in a tube 50 centimeters long 
rotates the plane of polarization of sodium light through 8.31 
degrees of angle. What is the amount of sugar in each cubic 
centimeter of solution? 



PROBLEMS. 589 

PROBLEMS. CHAPTER XXVI. 

PHOTOMETRY AND ILLUMINATION. 

393. The intensity of illumination at a distance of four feet 
from a i6-candle-power lamp is sufficient for reading ordinary 
book type. Express this intensity of illumination in lumens per 
square foot (foot-candles) and find the distance from a 20-candle- 
power lamp at which the lamp will give the same intensity of 
illumination. 

394. The glow lamp which is used as a standard in a Bunsen 
photometer has a candle-power of 16.8 candles in the direction 
towards the photometer screen. Another lamp B is placed at. 
the other end of the photometer bar; and when the screen is ad- 
justed to equality of illumination on both sides, it is 2.61 meters 
from the lamp B, and 1.80 meters from the standard lamp. 
What is the candle-power of B in the direction towards the 
screen? Ans. 35.3 candle-power. 

395. A 20-candle-power beam strikes a surface obliquely, the 
angle between the axis of the beam and the normal to the surface 
is 35°. Find the intensity of illumination of the surface in 
lumens per square foot. 

396. Let the brightness of daylight with the sun at the zenith 
be taken as unity. What is the brightness of daylight with the 
sun 10° above the horizon making no allowance for the light 
absorbed by the atmosphere? 

397. The spherical angle subtended by a lens as seen from a 
given point on the axis of the lens is very nearly equal to Aja^ 
w^here A is the area of the lens and a is the distance of the given 
point from the lens. A 60-candle-power lamp with a very small 
filament is placed at a distance of 50 centimeters from a lens 
which is 15 centimeters in diameter and 30 centimeters in focal 
length so that the light from the lamp is concentrated by the 
lens at a point which is 75 centimeters beyond the lens, (a) 
What is the spherical angle of the cone of rays which strikes the 
lens from the lamp? (h) How many lumens of light pass through 



590 APPENDIX A. 

the lens? (c) What is the spherical angle of the cone of rays 
beyond the lens? {d) What is the candle-power of the cone of 
rays beyond the lens on the assumption that no light is lost? 

398. A beam of parallel rays of light has a sectional intensity 
of 10 lumens per square foot. Find the conical intensity of the 
beam in candle-power after it passes through a lens of which the 
focal length is 2 feet. Ans. 40 candle-power. 

PROBLEMS. CHAPTER XXVIII. 

TONES AND NOISES. 

399. The disk of a siren is driven at a speed which brings its 
tone into unison with the tone of a tuning fork. The speed of 
the disk is then observed to be 1,820 revolutions per minute, and 
the row of holes in the siren disk contains 20 equidistant holes. 
What is the frequency of vibration of the tuning fork? Ans. 
606.7 vibrations per second. 

Note. The sound of a siren is very rich in overtones and it is sometimes very 
difficult to tell whether a given tone is in unison with the fundamental tone of the 
siren or one of its overtones. 

400. Two tones of which the frequencies are 275 and 300 vibra- 
tions per second respectively, are overtones of an unknown funda- 
mental. Of the various possible values of frequency of the fun- 
damental, what is the largest? Ans. 25 vibrations per second. 

PROBLEMS. CHAPTER XXIX. 

FREE VIBRATIONS OF ELASTIC BODIES. 

401. A string having a mass of o.i gram per centimeter is 
under a tension of 10,000,000 dynes. What is the velocity of a 
wave (a bend) on the string? 

402. What is the frequency of vibration of the string (i meter 
long) of problem 401 in its fundamental mode? 

403. A certain guitar string makes 400 vibrations per second. 
The length of the string is 36 inches and the second fret is 4 



PROBLEMS. 591 

Inches from the end of the string. How many vibrations per 
second does the string make when it is pushed down against the 
second fret? Ans. 450 vibrations per second. 

404. An organ pipe 10 feet long is open at both ends. What 
is the frequency of vibration of the fundamental tone of the pipe 
and what are the vibration frequencies of the respective overtones 
of the pipe? Ans. fundamental, 54.5 vibrations per second. 

405. An organ pipe 5 feet long is closed at one end. What is 
the vibration frequency of its fundamental tone and what are the 
vibration frequencies of the respective overtones of the pipe? 
Ans. fundamental, 54.5 vibrations per second. 

406. Sound travels 3.8 times as fast in hydrogen as in air. 
What would be the fundamental tone of an organ pipe 6 feet long, 
open at both ends and filled with hydrogen? Ans. 345.17 
vibrations per second. 

407. A horn has a tube 4 feet long. Calculate the length of 
the auxiliary tube required to lower its pitch in the ratio of 9 : 8. 
Ans. 0.5 foot. 

Note. The air column in a horn vibrates as in an organ pipe open at both ends. 

PROBLEMS. CHAPTER XXX. 

FORCED VIBRATIONS AND RESONANCE. 

408. A bass voice sings the vowel a as in father on a note of 
which the frequency is 100 vibrations per second. A soprano 
voice sings the same vowel on a note of which the frequency is 
700 vibrations per second. Make a diagram on which the 
overtones of both notes are laid down as points at distances 
proportional to their frequencies, and make a point in this dia- 
gram showing the frequency of the tone which characterizes 
the vowel. Which of the two singers produce the vowel most 
distinctly and why? 

409. What must be the relative proportions of air and hydrogen 
in the mouth cavity of a speaker to give a as in part when the 



592 



APPENDIX A. 



attempt is made to produce a as in paw? Ans. 7 parts hydrogen 
and 6 parts air by volume. 

Note. The frequency of vibration of the gas in an organ pipe or any cavity is 
inversely proportional to the square root of the density of the gas. The density of 
a mixture of a volumes of air and h volumes of hydrogen is to the density of air 
as a + ^/i4 is to a -\r h and the frequency of vibration of the mouth cavity when 
filled with the mixture is to its frequency when filled with air as Va + /f is to 
V^ {hji4). 



APPENDIX B. 

DIFFERENTIAL CALCULUS AND INTEGRAL CALCULUS. 

In the study of phenomena which depend upon conditions 
which vary in time, that is, upon conditions which change from 
instant to instant, it is necessary to direct the attention to what 
is taking place at an instant; or in other words to what takes 
place during a very short interval of time ; or, borrowing a phrase 
from the photographer, to make a snap shot, as it were, of the 
varying conditions. 

In the study of phenomena which depend upon conditions 
which vary from point to point in space, the attention must be 
directed to what takes place in a very small region. 

This paying attention to what takes place during a very short 
Interval of time or in a very small region of space does not refer 
to observation, but to thinking, it Is a mathematical method and 
it Is called calculus. 

The method of differential calculus. A phenomenon may be 
prescribed as a pure assumption, and the successive instantaneous 
aspects may be derived from this prescription. Thus In Art. 26 
the amount of water in a leaking pail is prescribed and the 
instantaneous rate of leak Is derived; the motion of a body in a 
circular path Is prescribed In Art. 31 and the instantaneous 
acceleration of the body is derived. 

Or a condition In space may be prescribed and the minute 
or point to point aspects may be derived from this prescription. 
Thus In Art. 28 the distribution of temperature along a rod is 
prescribed and the temperature gradient at a point is derived. 

Another aspect of the method of differential calculus is illus- 
trated by equation (24) of Art. 50. Recognizing the non- 
uniformity of pressure in a region we formulate the force action 
on an exposed surface by expressing the force per unit area on 

AF 
a very small portion of the surface, namely, — = p. 

39 593 



594 APPENDIX B. 

The method of integral calculus. It frequently happens that 
we recognize and can easily formulate what takes place during 
a very short time or in a very small region (as, for example, when 
we express the force action on a very small portion of an area 
which is exposed to the action of a fluid as in Art. 50). The 
problem then is to build up an expression for what takes place 
during a finite interval of time or throughout a finite region of 
space. This is the method of integral calculus. 

Any arithmetical argument which leads from a proposition 
concerning a rate or a gradient to a proposition concerning actual 
values of a varying quantity is an example of the method of 
integral calculus. In most cases this argument requires an 
elaborate use of algebra but the following is a very simple 
example: A man saves money at the rate of 75 cents a day, and 
it IS evident (we do not need to give the argument) that if he 
begins with nothing he has 75 dollars after 100 days. 

Detailed example of the method of integral calculus. Spin- 
inertia of a rotating disk. The identity of form of the equa- 
tions of translatory motion and the equations of rotatory motion 
shows that the kinetic energy of a rotating body must be express- 
ible thus 

W = ^Ks^ (i) 

where W is the kinetic energy of a body whose spin-inertia is K 
and whose spin-velocity is s. This equation corresponds to 
equation (22) of Art. 46 expressing the kinetic energy of a body 
in translatory motion. 

Consider a rotating disk of radius r rotating at spin-velocity 
s. Let A be the mass of the disk per unit area. Imagine the 
disk to grow larger by the addition of a very narrow strip of the 
same material to its rim. Let Ar be the width of this added 
strip, then 27rr-Ar is its area and 2irAr'Ar is its mass. 

Let AW he the increase of kinetic energy of the enlarged disk 
(the kinetic energy of the added material whose velocity is rs 
as explained in Art. 34). Then we have 

AW = ^{2irAr'Ar) X {rsY 



DIFFERENTIAL AND INTEGRAL CALCULUS. 595 

according to equation (22) of Art. 46. Therefore, reducing this 
expression we get 

AW = 2TAsh^-Ar (ii) 

This equation shows that the kinetic energy of the growing disk 
increases {2'jrAs^)r^ times as fast as r, because when r in- 
creases by an indefinitely small amount Ar the increase of kinetic 
energy is {2irAs^)r^ times as great. The expression in the pa- 
renthesis is a constant under the assumed conditions. 

Consider any quantity y which is given by the equation : 

y = br^ (iii) 

Then if r increases we have 



y + Ay = b(r -{- ArY 



or 



y-{-Ay = br^ + 4br''Ar + \^^^^^ ,^^^,_^ ,,^,, | (iv) 



terms contammg 
(Ar)2, {Ary and (ArY 



Therefore, subtracting equation (iii) from equation (iv) member 
by member and dividing both members of the resulting equation 
by Ar, we get 

— = ^br^ + terms containing (Ar), (Ar)^, and (ArY (v) 

This equation shows that the limiting value of the ratio Ay/Ar 
is equal to 4.br^. That is to say, br"^ is a quantity which grows 
4&r^ times as fast as r. Therefore writing 2TrAs'^ for 46 (or 
^ttAs'^ for b) we see that ^irAs^r^ is a quantity which grows 
2t A s^r^ times as fast as r just as W is known to grow by 
equation (ii). 

Any two quantities which have the same law of growth must have 
a constant difference; thus if John and Henry always save or spend 
money at the same rate, then J — H must be constant, where 
J is the amount of money John has and H is the amount of money 
Henry has. Or consider the two exactly similar curves A and 



596 



APPENDIX B. 



B in Fig. 479. These curves define the ordinates y and y' as 
functions of x, and as x increases these quantities y and y' 




increase and decrease together, that is to say, the two quantities 
or functions y and y' have exactly the same law of growth as x 
changes, and it is evident from the figure that the difference 
y' — y is constant in value (equal to C). 

Now it is shown above that the kinetic energy T^ of a rotating 
disk has exactly the same law of growth (due to an increase of 
radius r of the disk) as the function Jf^I^V, so that we must have 



or 



W — iTT^^V = a constant C 



W = Jtt^^V + a constant C 



(vi) 



But W must be zero when r is zero because a disk of zero radius 
has no kinetic energy. Therefore the unknown constant C 
must be zero, so that equation (vi) becomes 

W = Jtt^^V^ (vii) 

But irr^A is the mass m of the disk so that equation (vii) becomes 

W = ^mr^ X s^ (viii) 

Comparing this equation with equation (i) we get . 



K = hmr^ 



(ix) 



DIFFERENTIAL AND INTEGRAL CALCULUS. 597 

That is to say, the spin-inertia of a disk with respect to its axis 
of figure is equal to half the product of its mass by the square 
of its radius as given in the table on page 22. 

The beginner in the study of calculus would be greatly helped 
by reading Chapter I (pages 1-43) of the second edition of 
Franklin, MacNutt & Charles' Calculus, published by the 
authors, South Bethlehem, Pa. 



INDEX 



Abampere, definition of, 210 
Abbe eye-piece, the, 413 
Abbe's sine law, 401 
Abhenry, definition of the, 278 
Absolute temperature (see Kelvin tem- 
perature). 
Acceleration, definition of, 6 

of spin, 16 
Accommodation of the eye, 382 
Achromatic lens, the, 406, 428 
Achromatization of focal length or 
achromatization of magnification, 408 
of focal plane, 407 
Action and reaction, 1 1 
Active and inactive forces, 60 
Addition of forces, 27 
Adhesion and cohesion, 82 
Air columns, vibration of, 497-511 

thermometer, the, iii 
Alternating-current transformer, 266 
Alternator, the, 262 

the single-phase, 264 
the three-phase, 264 
the two-phase, 264 
Ammeter, the direct-current, 201 

multiplying shunt, 249 
Ampere, definition of, 210 

the international standard, 216 
Archimedes' principle, 79 
Armature windings, ring and drum, 
Arrester, the lightning, 284 
Anastigmatic lens, the, 398 
Anode, definition of, 215 
Aperture, numerical, of a lens, 394 
Aplanatic lens, 401 
Aplanatism, definition of, 401 
Apochromatic microscope object glass, 

the, 417 
Astigmatic pencil of rays, 357 
Astigmatism of the eye, 400 

of a lens, 398 
Atomic theory and mechanical theory, 
322 
and thermodynamics, 11 9-1 21 
of electricity. 322 
of gases, 325 
Attraction, electrostatic, 286 
Audion, the, 340 



Back electromotive force in motor 

armature, 252 
Balanced and unbalanced forces, 7 

torques, 17 
Ballistic galvanometer, the, 288 
Barometer, the, 77 
Base ball curves, 98 
Battery, electric, or voltaic cell, 216 

polarization of, 242 
Beats and combination tones, 526 
Bells, vibration of, 518 
Bernoulli's principle, 95 
Bipolar dynamos and multipolar dy- 
namos, 208 
Blow-out, the magnetic, 202 
Boiling points and melting points, 132 
Bourdon gauge, the, 79 
Boyle's law, 109 
Bright-line spectra, 425 
Brownian motion, the, 124 
Bunsen photometer, 468 
Buoyant force, 79 

Calculus, 593 

Calorimeter, the water, 127 

Canal rays and cathode rays, 334 

Candle-power, definition of, 463 

Candle, the British standard, 462 

Capacity, electrostatic, definition of, 290 

Capillary attraction, 82 

Carcel lamp, the, 462 

Carrying capacities of wires, 229 

Cathode, definition of, 215 

rays and canal rays, 334 
Caustic surface, definition of, 365 
Center of mass, 14 
Centrifugal drier, the, 43 
c.g.s. system of units, 26 
Chemical effect of the electric current, 

214 
Charge, electric (see electric charge). 
Charging by influence, 314 
Chladni's figures, 517 
Choke coil, 279 

Choking effect of inductance, 279 
Chromatic aberration of a lens, 405 
Circular motion, 38 
Cohesion and adhesion, 82 
Color, 475 



598 



INDEX 



599 



Color blindness or di-chroic vision, 481 
contrasts, 481 
mixing, 477 
sensation and brightness sensation, 

475 
the Young-Helmholtz theory of, 

479 
Colors, complementary, 479 

of thin plates and films, 432 
Coma, definition of, 397 

description of, 400 
Combination tones, 526 
Combined resistance of parallel branch- 
es, 250 
Compass, the magnetic, 186 
Complementary colors, 479 
Components of a force, 29 
Compound dynamo, the, 259 

microscope, 386 
Condenser, capacity of, 289 

the electric, 282 
Conduction of heat, 131 
Conductivity, thermal, definition of , 131 
Conductors and insulators, electric, 184 
Conjugate points of lenses, 374 
Connections in parallel, 248 

in series, 247 
Constant and variable quantities, 31 
Continuous and discontinuous variables, 

37 
spectra, 425 

Coolidge tube, the, 338 

Corona discharge, the, 319 

Cottrell process for precipitating dust 
and smoke, 319 

Coulomb, definition of the, 287 

Counter electromotive force (see back 
electromotive force). 

Critical states, 145 

Crookes tube, the, 332 

Current carrying capacities of wires, 229 
division of in parallel branches, 248 
strength magnetically defined, 209 

Curvature of field of lens, 404 

Curve, the easement, 42 

Cycle, definition of, 265 

Dalton's law, 145 

Damping, 518 

Dark-line spectra, 426 

Daniell cell, the, 222 

D'Arsonval galvanometer, the, 201 

Density and heaviness, 3 

and specific gravity, 81 
Dew point, definition of, 146 
Dichroic vision or color blindness, 481 
Dielectric strength, 297 
Differential calculus, 593 



Diffraction and interference of light, 431 

grating, the, 437-441 

of light, 436 

definition of, 356 
Diopter, definition of the, 383 
Direct-current ammeter, the, 201 
Direct- vision spectroscope, 427 
Discharge rate of a stream, 91 
Discontinuous and continuous variables, 

37 
Dispersion and spectrum analysis, 419 
Dissipation of energy, 123 
Distortion of image by lens, 402 
Double refraction, 446 

and polarization, 442 
Huygens' theory of, 449 
Doubler, the electric, 315 
Drier, the centrifugal, 43 
Drum and ring armature windings, 209 
Dry cell, the, 221 
Dynamics, 5 

Dynamo, the alternating-current (see 
alternator). 

the compound, 259 
the direct-current, 203, 207 
fundamental equation of the direct- 
current, 253 
the series, 259 
the shunt, 257 
Dynamos, bipolar and multipolar, 208 
Dyne, the, definition of, 9 
Dystectic points, 142 

Ear, the, 524 
Easement curve, the, 42 
Eddy currents, 274 
Effiux of a liquid from an orifice, 94 
Electrical screening, 310 
Electric battery or voltaic cell, 216 
charge, idea of, 283 
current strength magnetically de- 
fined, 209 

chemical effect of, 214 
heating effect of, 229 
doubler, 315 
field, intensity of, 306 

the, 304 
motor, the direct-current, 203 
Electrochemical equivalent, definition 

of, 224 
Electrodes, definition of, 214 
Electrolyte, definition of, 214 
Electrolytic cell, definition of, 214 
Electrolysis, 214 

the laws of, 225 
Electromagnet, the, 181 
Electro-mechanics and electro-atomics, 
322 



6oo 



INDEX 



Electrometer (see electrostatic volt- 
meter), 286 
Electroplating, 182 
Electromotive force, induced, 265 
definition of, 239 
drop in generator or along a 
line, 245 
Electron theory, the, 322 
Electrons and ions, 328 
Electroscope, the gold-leaf, 316 

the pith-ball, 309 
Electrostatic attraction, 286 

voltmeter, 286 
Endothermic reactions, 130 
Energy, definition of, 65 

the conservation of, 68 

the dissipation of, 123 
Engine, the steam, 163-169 
Entropy, general expression for change 
of during a reversible process, 172 

of an ideal gas, 171 

law of, 1 53-161 

of a substance, 169, 171, 172 
Erg, definition of the, 62 
Eutectic points and euteclectic mix- 
tures, 142 
Exothermic reactions, 130 
Expansion, coefficients of, 117 

free and constrained, 169 

thermal, of gases, 109 

of solids and liquids, 116 
Eye, the, 381 

accommodation of, 382 

imperfections of, 383 
Eye-pieces, examples of, 412-413 

Falling bodies, 23 

Faraday's "ice pail experiment" (see 

Art. 219)- 
Field angle of a lens, 395 

curvature of a lens, 404 

glass, the, 389 
First law of thermodynamics, 124 
Flame, the sensitive, 88 
Flow, lamellar, 87 

permanent and varying, 86 

simple, 86 
Fluoroscope, the, 337 
Flux, magnetic, definition of, 194 
Focal points and focal lengths of lenses, 

373 
Foot-candle, definition of, 465 
Foot-" pound," definition of the, 62 
Force addition, 27 

and its effects, 5 

components of, 29 

parallelogram, the, 27 

resolution of, 29 

units of, 9 



Forced vibrations and resonance, 518 
Forces, active and inactive, 60 

balanced and unbalanced, 7 
Foucault currents (see eddy currents). 
Fourier's theorem, 492 
f.p.s. system of units, 26 
Fraunhofer's lines, 426 
Free vibrations, 494 
Frequencies, standard, 265 
Frequency, definition of, 265 
Friction, coefficient of sliding, 59 

of fluids in pipes and channels, 102 
Fusion, latent heat of, 144 

Galvanometer, the ballistic, 288 

the D 'Arson val, 201 

the moving coil, 201 

the needle, 196 

the tangent, 212 
Gas engine ignition, 280 
Gay Lussac's law, 109 
Gee-pound, definition of, 11 
Geissler tube, the, 332 
Generator, the direct-current, 203, 208 
Glare, 473 

elimination of, 474 
Gold-leaf electroscope, 316 
Gradient, definition of, 35 

example of, 36 
Grating, the diffraction, 437-441 
Gyro-compass, the, 51 
Gyroscope, the, 46-50 

uses of, 50 

Harmonic motion, 53 
Heat and light, 460 

conductivity, definition of, 131 

measurement of, 126 

of combustion, 129 

of reaction, 129 

specific, definition of, 128 
Heating effect of the electric current, 

229 
Heaviness and density, 3 
Hefner lamp, the, 462 
Henry, definition of the, 278 
Hertz oscillator, the, 303 
Holmgren test for color blindness, 483 
Homocentric pencil of rays, 356 
Hooke's law and simple harmonic 

motion, 53 
Horse-power, definition of the, 63 
Horse-power-hour, definition of the, 64 
Humidity, absolute, definition of, 146 

relative, definition of, 147 
Huygens eye-piece, the, 412 
Huygens' construction, 355 

principle, 355 

application of, 359 



INDEX 



60 1 



Hydraulic ram, the, 87 
Hydraulics, 85 
Hydrostatic pressure, 71 
Hydrostatics, 71 
Hygrometry, 146 

Ignition, jump spark, 280 

wipe spark, 280 
Illumination of a room, 471 

and photometry, 460 
Image distortion, 402 

formation by lenses, geometry of, 

375 

of an object, 363 
Index of refraction, 359 
Induced electromotive force, 252-257, 

265 
Inductance, choking effect of, 279 

definition of, 278 

of a circuit, discussion of, 276 
Induction coil, the, 269 
Inductivity, definition of, 292 
Inertia, discussion of, 20 

spin, of a disk, 594 
Influence, charging by, 314 

machine, the, 317 
Infra-red rays, 420 
Integral calculus, 594 
International standard ampere, 216 
Insulators and conductors, electric, 184 
Interference and diffraction of light, 431 
Interferometer, Michelson's, 434 
Invar alloy, 118 

Ionization by the electric field, 330 
Ions and electrons, 328 

Jet pump, the, 97 

Joule, definition of the, 62 

Joule's law, 232 

Kelvin temperatures, 113, 115, 160, 166 
Kenetron, the, 340 
Kilowatt, definition of, 63 
Kilowatt-hour, definition of the, 64 
Kinetic energy, definition of, 66 
theory of gases, 325 

Lamination and eddy currents, 274 
Lamps, standard, 462 
Lantern, the projection, 381 
Latent heat, definition of, 144 
Laws of motion, 7-20 
Lens, aplanatic, 401 

astigmatism of, 398 

chromatic aberration of, 405 

equations, 378-380 

field angle of, 395 

field curvature of, 404 

imperfections, 392 



Lens, numerical aperture of, 394 

orthoscopic, 404 

rectilinear, 404 

spherical aberration of, 396 

systems, compensated, examples of, 
410-418 

the achromatic, 406, 428 

the ideal, simple, 373 

the wide-angle, 408 

with fiat field, 405 
Lenses, conjugate points of, 374 

focal points and focal lengths of, 

373 

image formation by, 374-375 

image formation, geometry of, 375 

simple, 371 
Light and radiant heat, 460 

and sound, transmission of, 349 

corpuscular theory of, 349 

definitions of, 348 

homogeneous and non-homogene- 
ous, 419 

monochromatic and polychromatic 
(see light, homogeneous and non- 
homogeneous). 

units, 463 

velocity of, 351 

wave theory of, 350 
Lightning arrester, the, 284 
Liquid air machine, the, 171 
Local action and voltaic action, 218 
Locomotive on curve, 41 
Loudness of tones, 488 
Lumen, definition of, 464 
Lux, definition of, 465 

Magnetic blow-out, the, 202 
field, intensity of, 191 

the, 190 
flux, definition of, 194 
Magnet pole, unit of, 188 
poles of, 187 

attraction and repulsion of, 189 
Magnifying glass, the, 384 

power of microscope, definition of, 

384 
Manometers, 78 
Mass and weight, 3 

center of, 14 

definition of, 4 
Maxwell, definition of the, 194 
Measures and units of quantity, 24 
Mechanical theory and atomic theory, 

322 
Melting points and boiling points, 132 
Mercury-in-glass thermometer, the, 113 
Michelsoii's interferometer, 434 
Microscope, magnifying power of, 384 

object glasses, examples of, 417 



602 



INDEX 



Microscope, the compound, 386 

the simple, 384 
Mirror, the plane, 361 

the spherical, 363-364 
Moment of inertia, see spin-inertia. 
Moments, the principle of, 17 
Motion, laws of, 7-20 

uniformly accelerated, 23 
Motor-generator, the, 269 
Motor-starting rheostat, the, 260 
Motor, the direct-current, 203 
Multiplying coil for voltmeter, 247 

shunt for ammeter, 249 
Multipolar dynamos, 208 

Newton's laws of motion, 7-20 
Nicol prism, the, 454 
Noises and tones, 487 

Ohm, definition of the, 233 

Ohm's law, 241 

Open-tube manometer, 78 

Opera glass, the, 389 

Optical instruments, simple, 381 

Organ pipes, 512 

Orthoscopic or rectilinear lens, 404 

Oscillations, electric, 300 

Oscillator, electric, the, 302 

Ozonizer, the, 320 

Parabolic reflector, the, 363 
Parallel branches, combined resistance 
of, 250 
division of current in, 248 

connections, 248 
Pascal's principle, 73 
Pencil of rays, definition of, 356 
Pendulum, the simple, 56 
Pentane lamp, the, 462 
Phases, coexistent, 107 
Phonograph, the, 523 
Photographic lenses, 413-417 
Photometer, the Bunsen, 468 
Photometric units, 463 
Photometry and Illumination, 460 

simple, 461 
Pitch, determination of, 489 

limits of audibility, 489 

of tones, 488 

standards of, 489 
Pi tot meter, the, loi 
Polariscope, the, 457 
Polarization and double refraction, 442 

of a battery, 242 

of light by reflection, 443 
Pole strength, magnetic, 188 
Poles of a magnet, 187 
Porro prisms, the, 389 
Poundal, the, definition of, 9 



Potential energy, definition of, 66-68 
Power, definition of, 62 

measurement of by ammeter and 
voltmeter, 245 

time units of work, 64 

units of, 63 
Precession, definition of, 540 
Prism, refraction of light by, 368 
Projection lantern, the, 381 

Quality or timbre of tones, 489 
Quantities, constant and variable, 31 

Radian of angle, explanation of, 16 

Radio-activity, 341 

Ramsden eye-piece, the, 413 

Ram, the hydraulic, 87 

Rate of change, definition of, 32 

Ray of light, definition of, 356 

Rays, anastigmatic pencil of, 357 

homocentric pencil of, 356 

pencil of, 356 
Reaction and action, 11 
Recalescence of steel, 149 
Receiver, the telephone, 273 
Rectifier, the vacuum-tube, 340 
Rectilinear or orthoscopic lens, 404 
Reflection and refraction, 359 

total, 368 
Reflector, the parabolic, 363 
Refraction and reflection, 359 

double, 442 

index of, 359 
Refractometer, the, 581 
Resistance, change of with temperature, 
236 

combined, of parallel branches, 250 

specific (see resistivity). 

the idea of, 231 
Resistivity, definition of, 234 
Resolution of forces, 29 
Resonance, 518 
Resonator, the, 520 
Rheostat, the, 233 
Ring and drum armature windings, 

209 
Rods, vibration of, 515 
Roentgen rays or X-rays, 337 
Rotary converter, the, 269 
Rotational and irrotational fluid mo- 
tion, 90 
Rotatory motion, 13 

Saccharimeter, the, 458 

Scalar and vector quantity, 26 

Screening, electrical, 310 

Second law of Thermodynamics, 153- 

161 
Sensation and stimulus, 347 



INDEX 



603 



Sensory nerves, 347 
Sensitive flame, the, 88 
Series connections, 247 

dynamo, the, 259 
Ship suction, 98 
Shunt, definition of, 248 
Shunt dynamo, the, 257 
Simple harmonic motion, 53 

microscope, the, 384 

voltaic cell, the, 217 
Slug, definition of, 11 
Snell's law of refraction, 367 
Sound and light, transmission of, 349 

definitions of, 348 

theory of, 485 

velocity of, 350 
Spark at break and water hammer, 281 

gauge, the, 298 
Specific gravity and density, 81 
definition of, 529 

heat, definition of, 128 

resistance (see resistivity). 
Spectra, bright-line, 425 

continuous, 425 

dark-line, 426 
Spectrometer, the, 426 
Spectroscope, the, 421-425 

the direct- vision, 427 

the grating, 439 
Spectrum analysts and dispersion, 419 
Spherical aberration, oblique (see astig- 
matism and coma), 
of a lens, 396 . 
axial, 397 

-candle, definition of, 463 

hefner, definition of, 463 

mirror, 363-364 
Spin acceleration, 16 
Spin-inertia, definition of, 21 
Spin-inertia, explanation of, 20 
of a disk, 594 
table of values of, 22 
Spin- velocity, 15 
Spit-ball, the, 89 
Spy-glass, the, 388 
Standard lamps, 462 
Standing wave trains, 501 
Starting rheostat, the motor, 260 
Steam engine, the, 163-169 
Steel, hardening and tempering of, 151 

recalescence of, 149 
Step-up transformation, 266 
Step-down transformation, 266 
Stigmatic lens, the, 398 
Stigmatic pencil of rays, 356 
Storage cell or battery, the, 223 
Stream lines, 86 
Strings, vibration of, 494-509 
Surface tension of liquids, 82 



Superheating and undercooling of 

liquids, 136 
Supersaturation of solutions, 136 
Systems of units, 26 

Tangent galvanometer, the, 212 
Telegraphy, wireless, 303 
Telephone receiver, the, 273 
repeater or amplifier, 340 
set, the, 273 
the, 271 

transmitter, the, 272 
Telescope object glasses, examples of, 
410-41 I 
the, 387-391 
Temperature, absolute (see Kelvin 
temperature). 

and thermal equilibrium, 107 
Temperatures, Kelvin, 113, 115, 160, 

166 
Thermal equilibrium and temperature, 

107 
Thermions, electrons from hot metal, 

337 

Thermodynamic degeneration, discus- 
sion of, 155 

Thermodynamics and the atomic the- 
ory, I I 9-1 2 I 
first law of, 124 
second law of, 153-161 

Thermometer, the air, iii 
the mercury-in-glass, 113 

Timbre or quality of tones, 489 

Time constant of a circuit, definition of, 
576 

Toepler-Holtz machine, the, 317 

Tone quality, 489 

Tones and noises, 487 

Torricelli's theorefn, 94 

Torque, definition of, 16 

Torques, balanced and unbalanced, 17 

Total reflection, 368 

Tourmaline crystals, action of, on light, 

443 
Transformation, step-down and step-up, 

266 
Transformer, the alternating current, 

266 
Translatory motion, 13 
Transmitter, the telephone, 272 
Tuning fork, the, 516 

Ultra-violet rays, 420 

Undercooling and superheating of 

liquids, 136 
Uniformly accelerated motion, 23 
Units and measures of quantity, 24 

of force, dynamic, 9 

systems of, 26 



6o4 



INDEX 



Vacuum-tube rectifier, 340 
Vapor pressures and temperatures, 133 
Vaporization, latent heat of, 144 
Variable and constant quantities, 31 
Variables, continuous and discontinu- 
ous, 37 
Vector addition, 27 

and scalar quantity, 26 
Velocity, constant and variable, 6 

of spin, 15 

variation of, 37 
Vibration of rods, 515 

of plates, 516 
Vibrations, forced, 518 

free, 494 
Visual angle, the, 383 
Volt, definition of, 243 
Voltage drop in generator or along a 

line, 245 
Voltaic action and local action, 218 

cell or electric battery, 216 

cells, types of, 220-223 
Voltmeter, electrostatic, 286 

multiplying coil, the, 247 

the, 244 



Vortex sheet, the, 88 
Vowel sounds, 521 

Water calorimeter, the, 127 

hammer and spark at break, 281 
vapor, pressures and temperatures 
of, 134 

Watt, definition of the, 63 

Watt-hour, definition of the, 64 

Wave-front, definition of. 353 

Wave-length, definition of, 353 

Wave pulses and wave trains, 353 
theory of light and sound, 350 

Wave-trains, advancing, 502 
standing, 501 

Weight and mass, 3 

Wide-angle lens, the, 408 

Wireless telegraphy, 303 

Wires, current carrying capacities of, 
229 

Work, definition of, 60 
units of, 62 

X-rays or Roentgen rays, 337 



